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53359611
Let \(u\) and \(v\) be polynomial functions, and define their product by \(w(x)=u(x)v(x)\). Suppose \(a\) is a real number. Decide whether this statement is always true, and justify your answer from the definition of \(w\): “If \(u(a)=0\), then \(w(a)=0\).”

Hints

- Translate “\(u\) has a zero at \(a\)” into an equation. - Substitute that equation into the product definition \(w(a)=u(a)v(a)\). - Focus on what happens when one factor of a product is zero.

Solution

1. By the definition of the product polynomial, \(w(a)=u(a)v(a)\). 2. If \(u(a)=0\), then \(w(a)=0\cdot v(a)=0\). 3. Therefore, the statement is always true.

Answer

True. If \(u(a)=0\), then \(w(a)=u(a)v(a)=0\).
55107311
Let \(A(x)=3x^3-2x+5\) and \(B(x)=-x^3+4x^2+7x-1\). Find and simplify \(A(x)+B(x)\).

Hints

- Line up terms that have the same power of \(x\). - A missing power in one polynomial has coefficient \(0\). - Combine coefficients without changing the exponents.

Solution

1. Add coefficients of like powers of \(x\). 2. The cubic terms give \(3x^3-x^3=2x^3\), the quadratic term is \(4x^2\), the linear terms give \(-2x+7x=5x\), and the constants give \(5-1=4\). 3. Therefore, \(A(x)+B(x)=2x^3+4x^2+5x+4\).

Answer

\(2x^3+4x^2+5x+4\)
55107511
Multiply and simplify: \(-3x^2(2x^3-x+4)\).

Hints

- Multiply the monomial by each term inside the parentheses. - Multiply coefficients and add exponents on like bases. - Check the sign of each product.

Solution

1. Distribute \(-3x^2\) to each term. 2. The products are \(-6x^5\), \(3x^3\), and \(-12x^2\). 3. Therefore, the simplified polynomial is \(-6x^5+3x^3-12x^2\).

Answer

\(-6x^5+3x^3-12x^2\)
55107811
Without expanding the entire product, determine the degree and leading coefficient of \(F(x)=(-3x^2+4x-1)(2x^4+x+5)\). Explain how you know.

Hints

- Focus only on the highest-degree term in each factor. - Degrees add when powers of \(x\) are multiplied. - The leading coefficient comes from multiplying the leading coefficients of the factors.

Solution

1. The highest-degree terms are \(-3x^2\) and \(2x^4\). 2. Their product is \(-6x^6\), and no other pair of terms can produce degree \(6\). 3. Therefore, \(F\) has degree \(6\) and leading coefficient \(-6\).

Answer

Degree \(6\); leading coefficient \(-6\).
53358711
The graph shows the functions \(f\), \(g\), and their sum \(h\), where \(h(x)=f(x)+g(x)\). Decide whether each statement is true or false. Justify your decisions. a) If \(f\) has a zero at some x-value, then \(h\) also has a zero there. b) At every x-value where the graphs of \(f\) and \(g\) intersect, \(h(x)\) is twice \(f(x)\). c) If \(f(x)=-g(x)\), then \(h\) has a zero at that x-value.
Figure for problem 533587

Hints

- Use a specific zero of \(f\) from the graph to test part a. - At an intersection, the two functions have equal outputs. - Substitute \(f(x)=-g(x)\) into the definition of \(h\).

Solution

a) False. For example, \(f(-2)=0\), but \(g(-2)=4\). Therefore, \(h(-2)=0+4=4\ne0\). b) True. At an intersection, \(f(x)=g(x)\). Therefore, \(h(x)=f(x)+g(x)=2f(x)\). c) True. If \(f(x)=-g(x)\), then \(h(x)=f(x)+g(x)=-g(x)+g(x)=0\).

Answer

a) False b) True c) True
53359411
Let \(f(x)=x^2\) and \(g(x)=2x\). Each panel shows the graphs of \(f\) and \(g\), along with one candidate graph labeled p or q. One candidate represents the sum \(s(x)=f(x)+g(x)\), and the other represents the difference \(d(x)=f(x)-g(x)\). Match p and q to the correct formulas. Justify your answer by testing a point or analyzing the zeros.
Figure for problem 533594

Hints

- Choose a simple input such as \(x=1\) and evaluate both combined functions. - Compare those outputs with the candidate graphs. - Alternatively, factor each combined polynomial and compare its zeros with the graphs.

Solution

1. Test \(x=1\): \(s(1)=1^2+2\cdot 1=3\), while \(d(1)=1^2-2\cdot 1=-1\). 2. Graph p passes through \((1, 3)\), so p represents \(s(x)=x^2+2x\). 3. Graph q passes through \((1, -1)\), so q represents \(d(x)=x^2-2x\). 4. The zeros confirm the matches: \(s(x)=x(x+2)\) has zeros \(-2\) and \(0\), while \(d(x)=x(x-2)\) has zeros \(0\) and \(2\).

Answer

p: \(s(x)=x^2+2x\); q: \(d(x)=x^2-2x\)
53359511
The panels show the graphs of \(f(x)=x^2\) and \(g(x)=x^4\). The graphs labeled p and q were formed by subtracting these functions. Which graph represents \(h(x)=f(x)-g(x)\), and which represents \(k(x)=g(x)-f(x)\)? Justify your answer by comparing \(x^2\) and \(x^4\) on \((-1, 1)\).
Figure for problem 533595

Hints

- Compare \(x^2\) and \(x^4\) when \(|x|<1\). - Determine the sign of each difference on that interval. - Notice that the two difference functions are opposites.

Solution

1. For \(-1<x<1\), \(x^4<x^2\) except at \(x=0\), where they are equal. 2. Therefore, \(h(x)=x^2-x^4\) is positive on \((-1, 0)\cup(0, 1)\) and equals zero at \(x=0\). Graph q has this behavior. 3. Also, \(k(x)=x^4-x^2=-h(x)\), so it is negative on \((-1, 0)\cup(0, 1)\). Graph p has this behavior.

Answer

p: \(k(x)=x^4-x^2\); q: \(h(x)=x^2-x^4\)
53359711
Let \(f\) and \(g\) be polynomial functions, and let \(h(x)=f(x)-g(x)\). Decide whether this statement is always true: “Whenever the graph of \(f\) is above the graph of \(g\), the corresponding value of \(h\) is positive.” Justify your answer algebraically.

Hints

- Translate the relative positions of the graphs into an inequality. - Subtract the same quantity from both sides. - Use the definition of \(h\).

Solution

1. If the graph of \(f\) is above the graph of \(g\) at an input \(x\), then \(f(x)>g(x)\). 2. Subtract \(g(x)\) from both sides to obtain \(f(x)-g(x)>0\). 3. Since \(h(x)=f(x)-g(x)\), it follows that \(h(x)>0\). Therefore, the statement is always true.

Answer

True. The condition \(f(x)>g(x)\) is equivalent to \(f(x)-g(x)>0\), so \(h(x)>0\).
55107411
Let \(P(x)=5x^4-3x^2+2x-7\) and \(Q(x)=2x^4+x^3-5x+4\). Find and simplify \(P(x)-Q(x)\).

Hints

- Treat subtraction as adding the opposite of the entire second polynomial. - Be especially careful with the signs of every term in \(Q(x)\). - Combine only terms with the same exponent.

Solution

1. Distribute the subtraction sign to every term of \(Q\): \(P(x)-Q(x)=5x^4-3x^2+2x-7-2x^4-x^3+5x-4\). 2. Combine like terms to get \(3x^4-x^3-3x^2+7x-11\).

Answer

\(3x^4-x^3-3x^2+7x-11\)
55107611
Multiply and write the result in standard form: \((2x-3)(x+5)\).

Hints

- Each term in one binomial must multiply each term in the other. - Keep the signs attached to their terms while distributing. - Combine the two linear terms after multiplying.

Solution

1. Distribute each term of the first binomial: \((2x-3)(x+5)=2x^2+10x-3x-15\). 2. Combine the linear terms: \(10x-3x=7x\). 3. The product is \(2x^2+7x-15\).

Answer

\(2x^2+7x-15\)
55107711
Multiply and write the result in standard form: \((x-2)(3x^2+5)\).

Hints

- Notice that the quadratic factor has no \(x\)-term. - Distribute each term of \(x-2\) across both terms of the quadratic. - Put the final terms in descending powers of \(x\).

Solution

1. Distribute \(x\): \(x(3x^2+5)=3x^3+5x\). 2. Distribute \(-2\): \(-2(3x^2+5)=-6x^2-10\). 3. Combine the results: \(3x^3-6x^2+5x-10\).

Answer

\(3x^3-6x^2+5x-10\)
55107911
Miguel simplifies \((4x^3-2x+5)-(x^3+6x-7)\) and writes \(3x^3+4x-2\). Identify Miguel's error and give the correct simplified polynomial.

Hints

- Track how the subtraction sign affects every term in the second polynomial. - Compare coefficients of like powers after the parentheses are removed. - Check whether the constant term also has the correct sign.

Solution

1. The subtraction sign must change the sign of every term in the second polynomial. 2. Rewrite the expression as \(4x^3-2x+5-x^3-6x+7\). 3. Combine like terms: \(3x^3-8x+12\). 4. Miguel's error was failing to distribute the negative sign correctly to all terms of the second polynomial.

Answer

The subtraction sign was not distributed to every term. The correct result is \(3x^3-8x+12\).
55578011
Let \(P(x)=x^2-3x+2\) and \(Q(x)=2x^3+x-5\). For each expression \(P+Q\), \(P-Q\), and \(PQ\): 1. State whether the result is a polynomial. 2. State its degree. You do not need to expand the product completely. Explain what the three results show about adding, subtracting, and multiplying polynomials.

Hints

- Compare the highest powers present before doing unnecessary expansion. - For a product, think about what happens to the leading terms. - After finding the three results, ask whether any operation introduced a non-integer or negative exponent of \(x\).

Solution

1. The sum \(P+Q\) is a polynomial. Its highest-degree term is \(2x^3\), so its degree is \(3\). 2. The difference \(P-Q\) is a polynomial. Its highest-degree term is \(-2x^3\), so its degree is \(3\). 3. The product \(PQ\) is a polynomial. The leading terms multiply to \(x^2\cdot 2x^3=2x^5\), so its degree is \(5\). 4. In each case, combining polynomial terms by addition, subtraction, or multiplication produces another polynomial. Thus, polynomials are closed under these three operations.

Answer

\(P+Q\): polynomial of degree \(3\) \(P-Q\): polynomial of degree \(3\) \(PQ\): polynomial of degree \(5\) Polynomials are closed under addition, subtraction, and multiplication.
52878111
The polynomial function \(f\) is given in factored form by \(f(x) = 3(x - 1)(x + 2)(x - 4)\). 1. Multiply the factors step by step and write the function in standard form \(f(x) = a_3x^3 + a_2x^2 + a_1x + a_0\). 2. Compare the value of \(a_3\) with the coefficient outside the factors. Explain why, in general, the leading coefficient of \(a_n(x - x_1)(x - x_2)\cdots(x - x_n)\) is \(a_n\).

Hints

- Use the distributive property to multiply two binomials first. - Multiply the resulting quadratic by the remaining binomial before applying the outside coefficient. - Identify how the highest-degree term is formed from the factors. - Track what the outside coefficient does to the leading term.

Solution

1. Multiply the first two factors: \((x - 1)(x + 2) = x^2 + x - 2\). 2. Multiply by the third factor: \((x^2 + x - 2)(x - 4) = x^3 - 3x^2 - 6x + 8\). 3. Multiply by the outside coefficient: \(f(x) = 3x^3 - 9x^2 - 18x + 24\). 4. Thus, \(a_3 = 3\), which equals the coefficient outside the product. 5. In a product of \(n\) monic linear factors, the term \(x^n\) is produced only by multiplying the \(x\)-term from every factor. Its coefficient is \(1\). Multiplying the entire product by \(a_n\) makes the leading term \(a_nx^n\), so the leading coefficient is \(a_n\).

Answer

1. \(f(x) = 3x^3 - 9x^2 - 18x + 24\) 2. \(a_3 = 3\). In general, multiplying the \(x\)-term from each monic linear factor produces \(x^n\), and the outside coefficient \(a_n\) makes the leading term \(a_nx^n\).
52878211
Consider the function \(p(x) = -2(x + 1)^2(x - 3)\). 1. Expand the expression and write the polynomial in standard form. 2. State the leading coefficient \(a_3\) and the constant term \(a_0\). 3. Explain how to identify the leading coefficient directly from this factored form without expanding the entire expression.

Hints

- Use the square of a binomial to expand \((x + 1)^2\). - The leading coefficient is the coefficient of the highest-degree term. - The constant term contains no variable. - To find only the leading term, multiply only the highest-degree term from each factor.

Solution

1. Expand the squared factor: \((x + 1)^2 = x^2 + 2x + 1\). 2. Multiply by the remaining factor: \((x^2 + 2x + 1)(x - 3) = x^3 - x^2 - 5x - 3\). 3. Multiply by \(-2\): \(p(x) = -2x^3 + 2x^2 + 10x + 6\). 4. Therefore, the leading coefficient is \(a_3 = -2\), and the constant term is \(a_0 = 6\). 5. Each linear factor is monic, so multiplying the leading terms gives \(x^2 \cdot x = x^3\). The outside coefficient \(-2\) therefore becomes the coefficient of \(x^3\).

Answer

1. \(p(x) = -2x^3 + 2x^2 + 10x + 6\) 2. \(a_3 = -2\) and \(a_0 = 6\) 3. Because the linear factors are monic, their leading terms multiply to \(x^3\); the outside coefficient \(-2\) is therefore the leading coefficient.
52878611
Consider the function \(f(x) = (x + 2)(x - 1)^2\). a) Expand the expression and write \(f(x)\) in standard polynomial form. b) State the degree and leading coefficient of the function. c) Find all zeros and state the multiplicity of each zero.

Hints

- Expand the squared binomial first. - Standard form lists polynomial terms in descending powers. - The degree and leading coefficient come from the highest-degree term. - The factored form makes the zeros easiest to identify. - The exponent on a factor gives the multiplicity of its zero.

Solution

1. Expand the squared factor: \((x - 1)^2 = x^2 - 2x + 1\). 2. Multiply by \(x + 2\): \((x + 2)(x^2 - 2x + 1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2\). 3. Combine like terms: \(f(x) = x^3 - 3x + 2\). 4. The highest exponent is \(3\), so the degree is \(3\). The coefficient of \(x^3\) is \(1\), so the leading coefficient is \(1\). 5. From the factored form, \(x + 2 = 0\) gives \(x = -2\), and \(x - 1 = 0\) gives \(x = 1\). 6. The factor \(x + 2\) occurs once, so \(x = -2\) has multiplicity 1. The factor \(x - 1\) is squared, so \(x = 1\) has multiplicity 2.

Answer

a) \(f(x) = x^3 - 3x + 2\) b) Degree \(3\); leading coefficient \(1\) c) \(x = -2\) with multiplicity 1; \(x = 1\) with multiplicity 2
53232411
A company has two divisions. Division A's monthly profit or loss, in thousands of dollars, is modeled by \(f(x)=-0.1x^2+0.8x+1\). Division B's monthly profit or loss is modeled by \(g(x)=0.5x-2\), where \(x\) is the month number. Define the company's total monthly profit by \(h(x)=f(x)+g(x)\). a) Write \(h(x)\) as a simplified polynomial. Then find the company's total profit in months \(x=2\) and \(x=8\). b) In which month is \(h(x)\) equal to Division A's profit \(f(x)\)? Explain your answer using \(g(x)\). c) Determine whether this statement is true or false, and justify your answer: “Whenever \(h(x)>0\), both divisions must have earned a profit.”

Hints

- Add the two polynomial expressions by combining like terms. - If \(f(x)+g(x)=f(x)\), determine what \(g(x)\) must equal. - To disprove the statement, find one month when the sum is positive but one function value is negative.

Solution

a) Add like terms: \(h(x)=(-0.1x^2+0.8x+1)+(0.5x-2)=-0.1x^2+1.3x-1\). Then \(h(2)=-0.4+2.6-1=1.2\), so the total profit is \(\$1200\). Also, \(h(8)=-6.4+10.4-1=3\), so the total profit is \(\$3000\). b) If \(h(x)=f(x)\), then \(f(x)+g(x)=f(x)\), which requires \(g(x)=0\). Solve \(0.5x-2=0\) to get \(x=4\). c) The statement is false. At \(x=2\), \(h(2)=1.2>0\), but \(g(2)=-1<0\). The company has a positive total profit even though Division B has a loss.

Answer

a) \(h(x)=-0.1x^2+1.3x-1\); month \(2\): \(\$1200\); month \(8\): \(\$3000\) b) Month \(4\), because \(g(4)=0\) c) False; for example, \(h(2)>0\) while \(g(2)<0\).
55108011
Polynomials \(A\) and \(B\) satisfy \(A(x)+B(x)=S(x)\), where \(A(x)=2x^3-x^2+4\) and \(S(x)=5x^3+3x^2-2x+9\). Find \(B(x)\) in standard form.

Hints

- Think of the equation as a missing-addend problem with polynomials. - Isolate \(B(x)\) before combining terms. - Distribute the subtraction sign to the entire known polynomial.

Solution

1. Rearrange the equation: \(B(x)=S(x)-A(x)\). 2. Substitute the given polynomials: \(B(x)=5x^3+3x^2-2x+9-(2x^3-x^2+4)\). 3. Combine like terms to obtain \(B(x)=3x^3+4x^2-2x+5\).

Answer

\(B(x)=3x^3+4x^2-2x+5\)
55108111
A custom-print shop models the revenue from an order of \(x\) shirts by \(R(x)=-2x^2+90x\) dollars and the cost by \(C(x)=20x+300\) dollars, for \(0\le x\le40\). a) Write the profit polynomial \(P(x)=R(x)-C(x)\) in standard form. b) Find \(P(20)\). c) Interpret \(P(20)\) in this context.

Hints

- Profit is revenue minus cost, so subtract the entire cost polynomial. - Simplify the polynomial before substituting the order size. - Interpret a positive profit value in dollars.

Solution

a) Subtract the cost polynomial: \(P(x)=(-2x^2+90x)-(20x+300)=-2x^2+70x-300\). b) Evaluate: \(P(20)=-2(20)^2+70(20)-300=300\). c) The model predicts a profit of \(\$300\) on an order of \(20\) shirts.

Answer

a) \(P(x)=-2x^2+70x-300\) b) \(P(20)=300\) c) The predicted profit for \(20\) shirts is \(\$300\).
55108211
The graph shows a polynomial function \(f\). Let \(g(x)=2x+3\) and define \(h(x)=f(x)+g(x)\). a) Use the graph to read \(f(-2)\), \(f(0)\), and \(f(2)\). Then find \(h(-2)\), \(h(0)\), and \(h(2)\). b) Use the graph to determine all shown x-values where \(h(x)=g(x)\). Explain the connection to \(f\).
Figure for problem 551082

Hints

- Read the y-value of \(f\) directly from the graph at each requested x-value. - Evaluate the algebraic function \(g\) separately before adding outputs. - For part b, simplify the equation \(f(x)+g(x)=g(x)\) before using the graph.

Solution

a) From the graph, \(f(-2)=3\), \(f(0)=-1\), and \(f(2)=3\). Also, \(g(-2)=-1\), \(g(0)=3\), and \(g(2)=7\). Therefore, \(h(-2)=2\), \(h(0)=2\), and \(h(2)=10\). b) Since \(h(x)=f(x)+g(x)\), the equation \(h(x)=g(x)\) holds exactly when \(f(x)=0\). The graph shows zeros of \(f\) at \(x=-1\) and \(x=1\).

Answer

a) \(f(-2)=3\), \(f(0)=-1\), \(f(2)=3\); \(h(-2)=2\), \(h(0)=2\), \(h(2)=10\) b) \(x=-1\) and \(x=1\)
55578111
Suppose \(P\) and \(Q\) are both degree-\(5\) polynomials. The leading coefficient of \(P\) is \(4\), and the leading coefficient of \(Q\) is \(-4\). a) Without expanding, determine the degree and leading coefficient of \(PQ\). b) Determine the degree and leading coefficient of \(P-Q\). c) What can you conclude about the degree of \(P+Q\)? Explain why its exact degree cannot be determined from the given information. d) Give one pair \(P,Q\) for which \(P+Q\) has degree \(4\), and another pair with the same stated degree and leading coefficients for which \(P+Q\) has degree \(2\).

Hints

- Track only the leading terms first for the product and difference. - In the sum, ask what happens when the two degree-\(5\) terms combine. - For part d), choose lower-degree terms deliberately so that different amounts of cancellation occur after the leading terms cancel.

Solution

1. The leading term of \(PQ\) is \((4x^5)(-4x^5)=-16x^{10}\). Therefore, \(PQ\) has degree \(10\) and leading coefficient \(-16\). 2. The leading term of \(P-Q\) is \(4x^5-(-4x^5)=8x^5\). Therefore, \(P-Q\) has degree \(5\) and leading coefficient \(8\). 3. In \(P+Q\), the degree-\(5\) terms cancel. Therefore, \(\deg(P+Q)<5\), but the exact degree depends on the lower-degree terms. 4. For example, let \(P_1(x)=4x^5+x^4\) and \(Q_1(x)=-4x^5+2x^4\). Then \(P_1+Q_1=3x^4\), which has degree \(4\). 5. Let \(P_2(x)=4x^5+x^4+x^2\) and \(Q_2(x)=-4x^5-x^4+2x^2\). Then \(P_2+Q_2=3x^2\), which has degree \(2\).

Answer

a) Degree \(10\), leading coefficient \(-16\) b) Degree \(5\), leading coefficient \(8\) c) \(\deg(P+Q)<5\), but its exact degree is not determined by the given information. d) One valid pair is \(P_1(x)=4x^5+x^4\), \(Q_1(x)=-4x^5+2x^4\). Another is \(P_2(x)=4x^5+x^4+x^2\), \(Q_2(x)=-4x^5-x^4+2x^2\).

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