Chapter 3

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-x^{2}+3 x+10$$

Find all real solutions. $$12 x^{3}=17 x^{2}+5 x$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(-2,1,\) and \(0 ; \quad P(-1)=-1\)

Answer each question. Which one of the following equations is set up for direct use of the zero- product property? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=3 x^{2}-2 x-6 ; \quad 1 \text { and } 2$$

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\sqrt{5} x^{3}+2 x^{2}-3 x+4$$

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}-6 x$$

Find all real solutions. $$3 x^{3}+3 x=10 x^{2}$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(2,5,\) and \(-3 ; \quad P(1)=-4\)

Answer each question. Only one of the following equations does not require Step 1 of the method for completing the square. Which one is it? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=-x^{3}-x^{2}+5 x+5 ; \quad 2 \text { and } 3$$

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-\sqrt{7} x^{3}-4 x^{2}+2 x-1$$

Solve each problem. Area of a Picture The mat around the picture shown measures \(x\) inches across. Which equation says that the area of the picture itself is 600 square inches? A. \(2(34-2 x)+2(21-2 x)=600\) $$\text { B. }(34-2 x)(21-2 x)=600$$ C. \((34-x)(21-x)=600\) D. \(x(34)(21)=600\) (IMAGE CAN'T COPY)

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}+4 x$$

Write each expression in standard form. Do not use a calculator. $$3 i+5 i$$

Find all real solutions. $$2 x^{3}+4=x(x+8)$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of 4 and \(1+i ; \quad P(2)=4\)

Answer each question. Only one of the following equations does not require Step 1 of the method for completing the square. Which one is it? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=2 x^{3}-8 x^{2}+x+16 ; \quad 2 \text { and } 2.5$$

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-\pi x^{5}+3 x^{2}-1$$

Solve each problem. Sum of Two Numbers Suppose that \(x\) represents one of two positive numbers whose sum is 30 (a) Represent the other of the two numbers in terms of \(x .\) (b) What are the restrictions on \(x ?\) (c) Determine a function \(P\) that represents the product of the two numbers. (d) Determine analytically and support graphically the two such numbers whose product is a maximum. What is this maximum product?

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-2 x-24$$

Write each expression in standard form. Do not use a calculator. $$5 i-(2-i)$$

Find all real solutions. $$3 x^{3}+18=x(2 x+27)$$

Answer each question. Only one of the following equations is set up so that the values of \(a, b,\) and \(c\) can be determined immediately. Which one is it? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(-7\) and \(2-i ; \quad P(1)=9\)

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=3 x^{3}+7 x^{2}-4 ; \quad \frac{1}{2} \text { and } 1$$

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\pi x^{7}-x^{5}+x-1$$

Solve each problem. Sum of Two Numbers Suppose that \(x\) represents one of two positive numbers whose sum is 45 (a) Represent the other of the two numbers in terms of \(x .\) (b) What are the restrictions on \(x ?\) (c) Determine a function \(P\) that represents the product of the two numbers. (d) For what two such numbers is the product equal to \(504 ?\) Determine analytically. (e) Determine analytically and support graphically the two such numbers whose product is a maximum. What is this maximum product?

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=3 x^{2}+3 x-6$$

Write each expression in standard form. Do not use a calculator. $$(-7 i)(1+i)$$

Find all complex solutions of each equation. $$7 x^{3}+x=0$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=16$$

For each polynomial, one or more zeros are given. Find all remaining zeros. \(P(x)=x^{3}-x^{2}-4 x-6 ; \quad 3\) is a zero.

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=2 x^{4}-4 x^{2}+3 x-6 ; \quad 1.5 \text { and } 2$$

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=2.74 x^{4}-3 x^{2}+x-2$$

Solve each problem. Maximizing Area A farmer has 1000 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enclosed by the fence?

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-2 x^{2}+6 x$$

Write each expression in standard form. Do not use a calculator. $$\frac{4+2 i}{i}$$

Find all complex solutions of each equation. $$2 x^{3}-4 x=0$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=144$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. \(P(x)=x^{3}-5 x^{2}+17 x-13 ; \quad 1\) is a zero.

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=x^{4}-4 x^{3}-x+1 ; \quad 0.3 \text { and } 1$$

Solve each problem. Maximizing Area A homeowner has 80 feet of fence mo enclose a rectangular garden. What dimensions for the garden give the maximum area?

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\sqrt{6} x^{6}-x^{5}+2 x-2$$

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-4 x^{2}+4 x$$

Determine whether each statement is true or false. If is false, tell why. Every real number is a complex number.

Find all complex solutions of each equation. $$3 x^{3}+2 x^{2}-3 x-2=0$$

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}=90$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. \(P(x)=x^{4}+2 x^{3}-10 x^{2}-18 x+9 ; \quad-3\) and 3 are Zeros.