Chapter 6

Fill in the blanks. To factor a polynomial means to express it as a _____ of two (or more) polynomials.

Integers that follow one another, such as 6 and \(7,\) are called ___________ integers.

Fill in the blanks. \(2 x^{2}+3 x-1=0\) and \(x^{2}-36=0\) are examples of ___equations.

Fill in the blanks. \(x^{2}+6 x+9\) is a _________ -square trinomial because it is the square of the binomial \(x+3\).

\(x^{3}+27\) is the ____ of two cubes and \(a^{3}-125\) is the difference of two ____.

Fill in the blanks. The trinomial \(x^{2}-x-12 _____ as the product of two binomials: \)(x-4)(x+3)$

The ____ coefficient of \(3 x^{2}-x-12\) is 3

Fill in the blanks. To ___________ a polynomial means to express it as a product of two (or more) polynomials.

Fill in the blanks. A polynomial is factored _____ when each factor is prime.

Fill in the blanks. A _____________________ triangle is a triangle that has a \(90^{\circ}\) angle.

Fill in the blanks. \(a x^{2}+b x+c=0\) is called the __ form of a quadratic equation.

Fill in the blanks. The binomial \(x^{2}-25\) is called a _________ of two squares and it factors as \((x+5)(x-5) .\) The binomial \(x^{2}+25\) is a ________ of two squares and since it does not factor using integers, it is ________ .

The factorization of \(x^{3}+8\) is \((x+2)\left(x^{2}-2 x+4\right) .\) The first factor is a binomial and the second is a _____.

Fill in the blanks. $A _____ trinomial cannot be factored by using only integers.

GCF stands for_______________. When we write \(2 x+4\) as \(2(x+2),\) we say that we have ______________ out the \(\mathrm{GCF}, 2\).

For each of the following polynomials, which factoring method would you use first? $$ 2 x^{5} y-4 x^{3} y $$

Fill in the blanks. The longest side of a right triangle is the _____________ . The remaining two sides are the _______________ of the triangle.

Fill in the blanks. Consider \(25 x^{2}+30 x+9\). a. The first term is the square of________. b. The last term is the square of ________. c. The middle term is twice the product of ________ and ________.

Fill in the blanks. The _____ coefficient of \(x^{2}-3 x+2\) is 1

Fill in the blanks. a. \(F^{3}+L^{3}=(\quad+\quad)\left(F^{2}-F L+L^{2}\right)\) b. \(F^{3}-L^{3}=(F \quad L)(\quad+F L+\quad)\)

The first terms of the binomial factors \((5 y+1)(y+3)\) are _____ and _____. The second terms of the binomial factors are _____ and _____.

Fill in the blanks. To factor \(m^{3}+3 m^{2}+4 m+12\) by __________ we begin by writing \(m^{2}(m+3)+4(m+3)\)

For each of the following polynomials, which factoring method would you use first? $$ 9 b^{2}+12 y-5 $$

Fill in the blanks. The ___________ theorem is a formula that relates the lengths of the three sides of a right triangle.

Fill in the blanks. Since the highest degree of any term in \(x^{3}-5 x^{2}-6 x=0\) is 3 it is called a ___ degree equation.

Fill in the blanks. Consider \(49 x^{2}-28 x y+4 y^{2}.\) The first term is the square of____. The last term is the square of____. The middle term is the opposite of twice the product of ____ and ____.

Fill in the blanks. A trinomial is factored _____ when no factor can be factored further.

Fill in the blanks. The terms \(x(x-1)\) and \(4(x-1)\) have the common __________ factor \(x-1\)

For each of the following polynomials, which factoring method would you use first? $$ x^{2}+18 x+81 $$

Fill in the blanks. a. \(x^{2}+2 x y+y^{2}=(\quad+\quad)^{2}\) b. \(x^{2}-2 x y+y^{2}=(x \quad y)^{2}\) \(-\quad)\) c. \(x^{2}-y^{2}=(x \quad y)(\quad-\quad)\)

Which of the following are quadratic equations? a. \(x^{2}+2 x-10=0\) b. \(2 x-10=0\) c. \(x^{2}=15 x\) d. \(x^{3}+x^{2}+2 x=0\)

Fill in the blanks. a. Before attempting to factor a trinomial, be sure that it is written in _____ powers of a variable. b. Before attempting to factor a trinomial into two binomials, always factor out any _____ factors first.

Complete each factorization. $$ \text { a. } 6 x=2 \cdot \quad \cdot x $$ $$ \text { b. } 35 h^{2}=5 \cdot \quad \cdot h $$ $$ \text { c. } 18 y^{3} z=2 \cdot \quad \cdot 3 \cdot \quad \cdot y \cdot \quad \cdot z $$

If \(10 x^{2}-27 x+5\) is to be factored as the product of two binomials, what are the possible first terms of the binomial factors?

For each of the following polynomials, which factoring method would you use first? $$ a x+a y-x-y $$

Write each equation in the standard form \(a x^{2}+b x+c=0\) a. \(x^{2}+2 x=6\) b. \(x^{2}=5 x\) c. \(3 x(x-8)=-9\) d. \(4 x^{2}=25\)

Fill in the blanks. a. \(36 x^{2}=(\quad)^{2}\) b. \(100 x^{4}=(\quad)^{2}\) c. \(4 x^{2}-9=(\quad)^{2}-(\quad)^{2}\)

Fill in the blanks. \(x^{2}+x-56=(x \square)(x \square\))

Fill in the blanks. a. \(x^{3}+64 y^{3}=(\quad)^{3}+(\quad)^{3}\) b. \(8 x^{3}-27=(\quad)^{3}-(\quad)^{3}\)

a. Find the GCF of \(30 x^{2}\) and \(105 x^{3}\). $$ \begin{array}{l} {30 x^{2}=2 \cdot 3 \cdot 5 \cdot x \cdot x} \\ {105 x^{3}=3 \cdot 5 \cdot 7 \cdot x \cdot x \cdot x} \end{array} $$ b. Find the GCF of \(12 a^{2} b^{2}, 15 a^{3} b,\) and \(75 a^{4} b^{2}\) $$ \begin{array}{l} {12 a^{2} b^{2}=2 \cdot 2 \cdot 3 \cdot a \cdot a \cdot b \cdot b} \\ {15 a^{3} b=3 \cdot 5 \cdot a \cdot a \cdot a \cdot b} \\ {75 a^{4} b^{2}=3 \cdot 5 \cdot 5 \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b} \end{array} $$

For each of the following polynomials, which factoring method would you use first? $$ x^{3}+27 $$

Multiply both sides of the equation by \(2 .\) Do not solve. $$ 10=\frac{1}{2} b(b+5) $$

List the squares of the integers from 1 through 20 .

Set \(5 x+4\) equal to 0 and solve for \(x\).

Fill in the blanks. \(x^{2}+5 x+3\) cannot be factored because we cannot find two integers whose product ____ is and whose sum is ____.

List the first ten positive integer cubes.

a. Fill in the blanks. When factoring a trinomial, we write it in _____ powers of the variable. Then we factor out any _____ (including \(-1\) if that is necessary to make the leading _____positive). b. What is the GCF of the terms of \(6 s^{4}+33 s^{3}+36 s^{2} ?\) c. Factor out \(-1\) from \(-2 d^{2}+19 d-8\)

a. Write a binomial such that the GCF of its terms is 2 . b. Write a trinomial such that the GCF of its terms is \(x\)

For each of the following polynomials, which factoring method would you use first? $$ y^{3}-64 $$

Fill in the blanks. a. If the length of the hypotenuse of a right triangle is \(c\) and the lengths of the other two legs are \(a\) and \(b\), then \(=c^{2}\) b. In a right triangle, the sum of the ____________ of the lengths of the two legs is equal to the square of the length of the ______