Chapter 6

Use multiplication to determine if each factorization is correct. a. \(9 y^{2}-12 y+4=(3 y-2)^{2}\) b. \(n^{2}-16=(n+8)(n-8)\)

What step should be performed first to solve \(x^{2}-6 x-16=0 ?\)

\((x-2)\left(x^{2}+2 x+4\right)\) is the factorization of what binomial?

Check to determine whether \((3 t-1)(5 t-6)\) is the correct factorization of \(15 t^{2}-19 t+6\).

Check to determine whether each factorization is correct. a. \(9 y^{3}+5 y^{2}-15 y=3 y\left(3 y^{2}+2 y-5\right)\) b. \(3 s^{3}+2 s^{2}+6 s+4=(3 s+2)\left(s^{2}+2\right)\)

For each of the following polynomials, which factoring method would you use first? $$ m^{2}+3 m n+2 n^{2} $$

What step (or steps) should be performed first before factoring is used to solve each equation? a. \(x^{2}+7 x=-6\) b. \(x(x+7)=3\)

Check to determine whether each factorization is correct. a. \(x^{2}-x-20=(x+5)(x-4)\) b. \(4 a^{2}+12 a-16=4(a-1)(a+4)\)

Use multiplication to determine if the factorization is correct. $$ b^{3}+27=(b+3)\left(b^{2}+3 b+9\right) $$

A trinomial has been partially factored. Complete each statement that describes the type of integers we should consider for the blanks. \(5 y^{2}-13 y+6=(5 y \square)(y \square)\) since the last term of the trinomial is positive and the middle term is negative, the integers must be _____ factors of 6.

Fill in the blanks to complete each factorization. $$ \begin{aligned} 2 x^{2}+6 x &=2 x \cdot x+2 x \cdot 3 \\ &= ( ) \end{aligned} $$

For each of the following polynomials, which factoring method would you use first? $$ 16-25 z^{2} $$

A ball is thrown into the air. Its height \(h\) in feet, \(t\) seconds after being released, is given by the formula \(h=-16 t^{2}+24 t+6\) When the ball hits the ground, what is the value of \(h ?\)

Check to determine whether the given number is a solution of the given quadratic equation. a. \(x^{2}-4 x=0 ; 4\) b. \(x^{2}-2 x-7=0 ;-2\)

Find two integers whose a. product is 10 and whose sum is 7 . b. product is 8 and whose sum is \(-6\) c. product is \(-6\) and whose sum is 1 d. product is \(-9\) and whose sum is \(-8 .\)

The factorization of \(y^{3}+27\) is \((y+3)\left(y^{2}-3 y+9\right) .\) Is this factored completely, or does \(y^{2}-3 y+9\) factor further?

A trinomial has been partially factored. Complete each statement that describes the type of integers we should consider for the blanks. \(5 y^{2}+13 y+6=(5 y \square)(y \square)\) since the last term of the trinomial is positive and the middle term is positive, the integers must be _____ factors of 6.

Fill in the blanks to complete each factorization. \(3 t^{3}-t^{2}+15 t-5=t^{2}(3 t-1)+5(3 t-1)\) = ( ) ( )

Complete each solution to solve the equation. $$ \begin{array}{rlrl} {(x-1)(x+7)} & {=0} & {} & {} \\ {x-1} & {\text { or }} & {} & {=0} \\ {x=1} & {} & {} & {x} & {=} \end{array} $$

Complete each factorization. $$ x^{2}-64=(x \quad 8)(x\quad 8) $$

Consider a trinomial of the form \(x^{2}+b x+c\) a. If \(c\) is positive, what can be said about the two integers that should be chosen for the factorization? b. If \(c\) is negative, what can be said about the two integers that should be chosen for the factorization?

A trinomial has been partially factored. Complete each statement that describes the type of integers we should consider for the blanks. \(5 y^{2}-7 y-6=(5 y \square)(y \square)\) since the last term of the trinomial is negative, the signs of the integers will be _____.

Consider the polynomial \(2 k-8+h k-4 h\) a. How many terms does the polynomial have? b. Is there a common factor of all the terms, other than \(1 ?\) c. What is the GCF of the first two terms and what is the GCF of the last two terms?

Use multiplication to determine whether the factorization is correct. $$ 5 c^{3} d^{2}-40 c^{2} d^{3}+35 c d^{4}=5 c d^{2}(c-7 d)(c-d) $$

Complete the solution to solve the equation. Fill in the blanks. a. Consecutive integers can be represented by \(x\) and___________ b. Consecutive odd integers can be represented by \(x\) and_________ c. Consecutive even integers can be represented by \(x\) and____________

Complete each factorization. \(16 t^{2}-49=(4 t+\quad)(4 t-\quad)\)

Fill in each blank to explain how to factor \(x^{2}+7 x+10\) by grouping. We express the middle term, \(7 x,\) as the sum of _____ terms: $$x^{2}+7 x+10=x^{2}+\square x+\square x+10$$

A trinomial has been partially factored. Complete each statement that describes the type of integers we should consider for the blanks. \(5 y^{2}+7 y-6=(5 y \square)(y \square)\) since the last term of the trinomial is negative, the signs of the integers will be_____.

What is the first step in factoring \(8 y^{2}-16 y z-6 y+12 z ?\)

Complete each factorization. $$ \begin{aligned} 6 m^{3}-28 m^{2}+16 m &=2 m\left(3 m^{2}-\square+8\right) \\ &=2 m(3 m-2)(\square-4) \end{aligned} $$

Geometry Problems Flags. The length of the flag of Australia is twice as long as the width. Find the dimensions of an Australian flag if its area is \(18 \mathrm{ft}^{2}\) (IMAGE CANNOT COPY)

Complete each solution to solve the equation. $$ \begin{aligned} &\begin{array}{r} {p^{2}-p-6=0} \\ {(-3)(p+2)=0} \end{array}\\\ &=0 \quad \text { or } \quad p+2=\\\ &p=\ \quad | \quad p= \end{aligned} $$

Determine whether each of the following is a perfect-square trinomial. $$ x^{2}+18 x+81 $$

Complete each factorization. $$ b^{3}+27=(\quad)\left(b^{2}-3 b+9\right) $$

Complete each factorization. $$ \begin{aligned} 2 a^{3}+& 3 a^{2}-2 a-3 \\ &=\square(2 a+3)-1(\square+3) \\ &=(\square)\left(a^{2}-1\right) \\ &=(2 a+3)(a+1)(\square) \end{aligned} $$

To factor a trinomial by grouping, a student made a table and circled the correct pair of integers, as shown. Enter the correct coefficients for the first stage in the factorization process. Key number \(=16\) $$\begin{array}{|c|c|}\hline \text { Factors } & {\text { Sum }} \\\\\hline 1 \cdot 16 & {17} \\\\\hline(2 \cdot 8& {10} \\\\\hline 4 \cdot 4 & {8} \\\\\hline\end{array}$$ $$x^{2}+\quad x+\quad x+16$$

Geometry Problems Billiards. Pool tables are rectangular, and their length is twice the width. Find the dimensions of a pool table if it occupies \(50 \mathrm{ft}^{2}\) of floor space.

Determine whether each of the following is a perfect-square trinomial. $$ x^{2}+14 x+49 $$

Complete each factorization. $$ z^{3}-125=(z-5)(\quad+5 z+\quad) $$

Complete each factorization. $$ 10 a^{4}-15 a^{3}=5 a \quad(2 a-3) $$

Factor. See Example 1 or Objective 1 $$x^{2}+3 x+2$$

Geometry Problems X-Rays. A rectangular-shaped x-ray film has an area of 80 square inches. The length is 2 inches longer than the width. Find its width and length. (IMAGE CANNOT COPY)

Solve each equation. $$ (x-3)(x-2)=0 $$

Determine whether each of the following is a perfect-square trinomial. $$ y^{2}+2 y+4 $$

Give an example of each type of expression. a. the sum of two cubes b. the cube of a sum

a. Suppose we wish to factor \(12 b^{2}+20 b-9\) by grouping. Identify \(a, b,\) and \(c\) b. What is the key number, \(a c ?\)

Factor. See Example 1 or Objective 1 $$ y^{2}+4 y+3 $$

Solve each equation. $$ (x+2)(x+3)=0 $$

Determine whether each of the following is a perfect-square trinomial. $$ y^{2}+4 y+16 $$

Give an example of each type of expression. a. the difference of two cubes b. the cube of a difference