Chapter 6

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$2 b x^{2}+44 b x+242 b$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Write a quadratic equation in standard form whose Solutions are \(-3\) and 5

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$y^{4}-\frac{y}{1000}$$

In factoring \(3 x^{2}-10 x-8,\) a student lists \((3 x-2)(x+4)\) as a possible factorization. Use FOIL multiplication to determine if this factorization is correct. If it is not correct, describe how the correct factorization can quickly be obtained using these factors.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x z^{2}-72 x z+432 x$$

Solve each equation. $$x^{3}-x^{2}-16 x+16=0$$

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$y^{4}-\frac{y}{8}$$

Explain why \(2 x-10\) cannot be one of the factors in the correct factorization of \(6 x^{2}-19 x+10\).

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$15 a^{2}+11 a b-14 b^{2}$$

Solve each equation. $$3^{x^{2}-9 x+20}=1$$

Factor \(x^{3}+3 x^{2}+2 x\). If \(x\) represents an integer, use the factorization to describe what the trinomial represents.

What is factoring?

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$0.25 x-x^{3}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm often able to use an incorrect factorization to lead me to the correct factorization.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$25 a^{2}+25 a b+6 b^{2}$$

Solve each equation. $$\left(x^{2}-5 x+5\right)^{3}=1$$

A box with no top is to be made from an 8-inch by 6 -inch piece of metal by cutting identical squares from each corner and turning up the sides (see the figure). The volume of the box is modeled by the polynomial \(4 x^{3}-28 x^{2}+48 x .\) Factor the polynomial completely. Then use the dimensions given on the box and show that its volume is equivalent to the factorization that you obtain.

What is a prime polynomial?

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$0.64 x-x^{3}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. First factoring out the GCF makes it easier for me to determine how to factor the remaining factor, assuming it is not prime.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$-36 x^{3} y+62 x^{2} y^{2}-12 x y^{3}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the trinomial correctly and then use your graphing utility to verify the factorization. \(x^{2}-5 x+6=(x-2)(x-3)\)

Explain how to find the greatest common factor of a list of terms. Give an example with your explanation.

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$(x+1)^{2}-25$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm working with a polynomial that has a GCF other than \(1,\) but then it doesn't factor further, so the polynomial that I'm working with is prime.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$-10 a^{4} b^{2}+15 a^{3} b^{3}+25 a^{2} b^{4}$$

Use an example and explain how to factor out the greatest common factor of a polynomial.

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$(x+2)^{2}-49$$

My graphing calculator showed the same graphs for \(y_{1}=4 x^{2}-20 x+24 \quad\) and \(\quad y_{2}=4\left(x^{2}-5 x+6\right), \quad\) so I can conclude that the complete factorization of \(4 x^{2}-20 x+24\) is \(4\left(x^{2}-5 x+6\right)\).

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$a^{2} y-b^{2} y-a^{2} x+b^{2} x$$

Suppose that a polynomial contains four terms and can be factored by grouping. Explain how to obtain the factorization.

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) Divide \(x^{3}-x^{2}-5 x-3\) by \(x-3 .\) Use the quotient to factor \(x^{3}-x^{2}-5 x-3\) completely.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$b x^{2}-4 b+a x^{2}-4 a$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the trinomial correctly and then use your graphing utility to verify the factorization. \(2 x^{2}+8 x+6=(x+3)(x+1)\)

Write a sentence that uses the word "factor" as a noun. Then write a sentence that uses the word "factor" as a verb.

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) Divide \(x^{3}+4 x^{2}-3 x-18\) by \(x-2 .\) Use the quotient to factor \(x^{3}+4 x^{2}-3 x-18\) completely.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$9 a x^{3}+15 a x^{2}-14 a x$$

Intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=x^{2}+3 x-4\) to solve $$x^{2}+3 x-4=0$$

Solve: \(4(x-2)=3 x+5 .\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. After factoring \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x,\) I noticed that I factored the monomial \(20 x^{3}\) in two different ways.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$4 a y^{3}-12 a y^{2}+9 a y$$

Intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=x^{2}+x-6\) to solve $$x^{2}+x-6=0$$

Graph: \(6 x-5 y \leq 30\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The trinomial \(3 x^{2}+2 x+1\) has relatively small coefficients and therefore can be factored.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$2 x^{4}+6 x^{3} y+2 x^{2} y^{2}$$

Intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=(x-2)(x+3)-6\) to solve $$(x-2)(x+3)-6=0$$

Graph: \(y=-\frac{1}{2} x+2 .\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The word greatest in greatest common factor is helpful because it tells me to look for the greatest power of a variable appearing in all terms.

Find all integers \(b\) so that the trinomial can be factored. $$3 x^{2}+b x+2$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x^{4}-9 x^{3} y+3 x^{2} y^{2}$$