Chapter 6

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 \(y=x^{2}-2 x+1\) to solve $$x^{2}-2 x+1=0$$

Multiply: \((2 x+3)(x-2)\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. You grouped the polynomial's terms using different groupings than I did, yet we both obtained the same factorization.

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

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$81 x^{4} y-y^{5}$$

Multiply: \((3 x+4)(3 x+1)\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. since the GCF of \(9 x^{3}+6 x^{2}+3 x\) is \(3 x,\) it is not necessary to write the 1 when \(3 x\) is factored from the last term.

Explain how to factor the difference of two squares. Provide an example with your explanation.

Find all integers \(b\) so that the trinomial can be factored. $$\text { Factor: } 3 x^{10}-4 x^{5}-15$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 x^{4} y-y^{5}$$

Factor by grouping: \(8 x^{2}-2 x-20 x+5\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\begin{aligned} a(x-7)+b(7-x) &=a(x-7)+b(-1)(x-7) \\ &=a(x-7)-b(x-7) \\ &=(x-7)(a-b) \end{aligned}$$

What is a perfect square trinomial and how is it factored?

Find all integers \(b\) so that the trinomial can be factored. $$\text { Factor: } 2 x^{2 n}-7 x^{n}-4$$

Factor completely. $$10 x^{2}(x+1)-7 x(x+1)-6(x+1)$$

Graph: \(y>-\frac{2}{3} x+1 .\) (Section 3.6, Example 3)

Explain why \(x^{2}-1\) is factorable, but \(x^{2}+1\) is not.

Solve the system: $$\left\\{\begin{array}{c}4 x-y=105 \\\x+7 y=-10\end{array}\right.$$

Factor completely. $$12 x^{2}(x-1)-4 x(x-1)-5(x-1)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(-4 x^{2}+12 x\) can be factored as \(-4 x(x-3)\) or \(4 x(-x+3)\)

Explain how to factor \(x^{3}+1\)

Write 0.00086 in scientific notation.

Factor completely. $$6 x^{4}+35 x^{2}-6$$

Solve: \(5 x+28=6-6 x\). (Section \(2.2,\) Example 7 )

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Suppose you receive \(x\) dollars in January. Each month thereafter, you receive \(\$ 100\) more than you received the month before. Write a factored polynomial that describes the total dollar amount you receive from January through April.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I factored \(9 x^{2}-36\) completely and obtained $$(3 x+6)(3 x-6)$$

Solve: \(8 x-\frac{x}{6}=\frac{1}{6}-8\)

Factor completely. $$7 x^{4}+34 x^{2}-5$$

will help you prepare for the material covered in the first section of the next chapter. Evaluate \(\frac{250 x}{100-x}\) for \(x=60\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Although I can factor the difference of squares and perfect square trinomials using trial-and-error, recognizing these special forms shortens the process.

Perform the indicated operation. $$(9 x+10)(9 x-10)$$

Factor completely. $$(x-7)^{2}-4 a^{2}$$

will help you prepare for the material covered in the first section of the next chapter. Why is \(\frac{6 x+12}{7 x-28}\) undefined for \(x=4 ?\)

Write a polynomial that fits the given description. Do not use a polynomial that appears in this section or in the Exercise Set. The polynomial has four terms and can be factored by grouping.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I factored \(9-25 x^{2}\) as \((3+5 x)(3-5 x)\) and then applied the commutative property to rewrite the factorization as \((5 x+3)(5 x-3)\)

Perform the indicated operation. $$(4 x+5 y)^{2}$$

Factor completely. $$(x-6)^{2}-9 a^{2}$$

will help you prepare for the material covered in the first section of the next chapter. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$\frac{x^{2}+6 x+5}{x^{2}-25}$$

Use a graphing utility to graph each side of the equation in the same viewing rectangle. Do the graphs coincide? If so, this means that the polynomial on the left side has been factored correctly. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$-3 x-6=-3(x-2)$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I compared the factorization for the sum of cubes with the factorization for the difference of cubes and noticed that the only difference between them is the positive and negative signs.

Perform the indicated operation. $$(x+2)\left(x^{2}-2 x+4\right)$$

Factor completely. $$x^{2}+8 x+16-25 a^{2}$$

Use a graphing utility to graph each side of the equation in the same viewing rectangle. Do the graphs coincide? If so, this means that the polynomial on the left side has been factored correctly. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{2}-2 x+5 x-10=(x-2)(x-5)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\begin{aligned}&\text { Because } x^{2}-25=(x+5)(x-5), \text { then } x^{2}+25=\\\&(x-5)(x+5)\end{aligned}$$

Factor completely. $$x^{2}+14 x+49-16 a^{2}$$

Use a graphing utility to graph each side of the equation in the same viewing rectangle. Do the graphs coincide? If so, this means that the polynomial on the left side has been factored correctly. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{2}+2 x+x+2=x(x+2)+1$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. All perfect square trinomials are squares of binomials.

Factor completely. $$y^{7}+y$$

Multiply: \((x+7)(x+10) .\) (Section 5.3, Example 1)

Factor completely. $$(y+1)^{3}+1$$