Chapter 6

Factor by grouping. $$x y+6 x+2 y+12$$

Factor completely. $$15 x^{5}-2 x^{4}-x^{3}$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 y^{5}-2 y^{2}$$

Factor completely. $$x^{2}+0.5 x+0.06$$

Factor using the formula for the sum or difference of two cubes. $$x^{3}+1$$

Factor by grouping. $$x y-x+5 y-5$$

Factor completely. $$6 x^{2}-3 x y-18 y^{2}$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 y^{5}-128 y^{2}$$

The formula $$N=\frac{t^{2}-t}{2}$$ describes the number of football games, \(N\), that must be played in a league with \(t\) teams if each team is to play every other team once. Use this information to solve If a league has 36 games scheduled, how many teams belong to the league, assuming that each team plays every other team once?

Factor completely. $$x^{2}-0.5 x-0.06$$

Factor using the formula for the sum or difference of two cubes. $$x^{3}+64$$

Factor by grouping. $$x y-x+7 y-7$$

Factor completely. $$4 x^{2}+14 x y+10 y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$6 x^{2}+8 x y$$

The formula $$N=\frac{t^{2}-t}{2}$$ describes the number of football games, \(N\), that must be played in a league with \(t\) teams if each team is to play every other team once. Use this information to solve If a league has 45 games scheduled, how many teams belong to the league, assuming that each team plays every other team once?

Factor completely. $$x^{2}-\frac{2}{5} x+\frac{1}{25}$$

Factor using the formula for the sum or difference of two cubes. $$x^{3}-27$$

Factor by grouping. $$3 x^{2}-6 x y+5 x y-10 y^{2}$$

Factor completely. $$12 x^{2}+10 x y-8 y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$21 x^{2}-35 x y$$

Factor completely. $$x^{2}+\frac{2}{3} x+\frac{1}{9}$$

Factor using the formula for the sum or difference of two cubes. $$x^{3}-64$$

Factor by grouping. $$10 x^{2}-12 x y+35 x y-42 y^{2}$$

Factor completely. $$24 x^{2}+3 x y-27 y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x y-7 x+3 y-21$$

You dive directly upward from a board that is 32 feet high. After \(t\) seconds, your height above the water is described by the polynomial$$-16 t^{2}+16 t+32$$ a. Factor the polynomial completely. b. Evaluate both the original polynomial and its factored form for \(t=2 .\) Do you get the same answer for each evaluation? Describe what this answer means.

Factor using the formula for the sum or difference of two cubes. $$8 y^{3}-1$$

Factor by grouping. $$3 x^{3}-2 x^{2}-6 x+4$$

Factor completely. $$8 x^{2} y+34 x y-84 y$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x y-5 x+2 y-10$$

You dive directly upward from a board that is 48 feet high. After \(t\) seconds, your height above the water is described by the polynomial$$-16 t^{2}+32 t+48$$ a. Factor the polynomial completely. b. Evaluate both the original polynomial and its factored form for \(t=3 .\) Do you get the same answer for each evaluation? Describe what this answer means.

Factor using the formula for the sum or difference of two cubes. $$27 y^{3}-1$$

Factor by grouping. $$4 x^{3}-x^{2}-12 x+3$$

Factor completely. $$6 x^{2} y-2 x y-60 y$$

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

Explain how to factor \(x^{2}+8 x+15\)

Factor using the formula for the sum or difference of two cubes. $$27 x^{3}+8$$

Factor by grouping. $$x^{2}-a x-b x+a b$$

Factor completely. $$12 a^{2} b-46 a b^{2}+14 b^{3}$$

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

Give two helpful suggestions for factoring \(x^{2}-5 x+6\)

Factor using the formula for the sum or difference of two cubes. $$125 x^{3}+8$$

Factor by grouping. $$x^{2}+a x+b x+a b$$

Factor completely. $$12 a^{2} b-34 a b^{2}+14 b^{3}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$72 a^{3} b^{2}+12 a^{2}-24 a^{4} b^{2}$$

In factoring \(x^{2}+b x+c,\) describe how the last terms in each factor are related to \(b\) and \(c .\)

Factor using the formula for the sum or difference of two cubes. $$x^{3} y^{3}-64$$

Factor each polynomial. $$24 x^{3} y^{3} z^{3}+30 x^{2} y^{2} z+18 x^{2} y z^{2}$$

Factor completely. $$-32 x^{2} y^{4}+20 x y^{4}+12 y^{4}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$24 a^{4} b+60 a^{3} b^{2}+150 a^{2} b^{3}$$