Chapter 6

What is a quadratic equation?

Without actually factoring and without multiplying the given factors, explain why the following factorization is not correct: $$x^{2}+46 x+513=(x-27)(x-19)$$

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

Factor each polynomial. $$16 x^{2} y^{2} z^{2}+32 x^{2} y z^{2}+24 x^{2} y z$$

Factor completely. $$-10 x^{2} y^{4}+14 x y^{4}+12 y^{4}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 a^{2}+27 a b+54 b^{2}$$

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Factor using the formula for the sum or difference of two cubes. $$27 y^{4}+8 y$$

Factor each polynomial. $$x^{3}-4+3 x^{3} y-12 y$$

Factor completely. $$30(y+1) x^{2}+10(y+1) x-20(y+1)$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 a^{2}+15 a b+18 b^{2}$$

If \((x+2)(x-4)=0\) indicates that \(x+2=0\) or \(x-4=0,\) explain why \((x+2)(x-4)=6\) does not mean \(x+2=6\) or \(x-4=6 .\) Could we solve the equation using \(x+2=3\) and \(x-4=2\) because \(3 \cdot 2=6 ?\)

Factor each polynomial. $$x^{3}-5+2 x^{3} y-10 y$$

Factor completely. $$6(y+1) x^{2}+33(y+1) x+15(y+1)$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$48 x^{4} y-3 x^{2} y$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. It's easy to factor \(x^{2}+x+1\) because of the relatively small numbers for the constant term and the coefficient of \(x\)

Factor each polynomial. $$4 x^{5}(x+1)-6 x^{3}(x+1)-8 x^{2}(x+1)$$

a. Factor \(2 x^{2}-5 x-3\) b. Use the factorization in part (a) to factor $$2(y+1)^{2}-5(y+1)-3$$ Then simplify each factor.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 a^{3} b^{2}-4 a b^{2}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I factor \(x^{2}+b x+c\) by finding two numbers that have a product of \(c\) and a sum of \(b\)

Factor each polynomial. $$8 x^{5}(x+2)-10 x^{3}(x+2)-2 x^{2}(x+2)$$

Factor using the formula for the sum or difference of two cubes. $$128-250 y^{3}$$

a. Factor \(3 x^{2}+5 x-2\) b. Use the factorization in part (a) to factor$$3(y+1)^{2}+5(y+1)-2$$ Then simplify each factor.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 a^{2}-32 a b+12 b^{2}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because some trinomials are prime, some quadratic equations cannot be solved by factoring.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. One factor of \(x^{2}+x+20\) is \(x+5\)

Factor each polynomial. $$3 x^{5}-3 x^{4}+x^{3}-x^{2}+5 x-5$$

Factor using the formula for the sum or difference of two cubes. $$64 x^{3}+27 y^{3}$$

Divide \(3 x^{3}-11 x^{2}+12 x-4\) by \(x-2\) Use the quotient to factor \(3 x^{3}-11 x^{2}+12 x-4\) completely.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 a^{2}-32 a b+12 b^{2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A trinomial can never have two identical factors.

Factor each polynomial. $$7 x^{5}-7 x^{4}+x^{3}-x^{2}+3 x-3$$

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

Divide \(2 x^{3}+x^{2}-13 x+6\) by \(x-2\) Use the quotient to factor \(2 x^{3}+x^{2}-13 x+6\) completely.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$7 x^{5} y-7 x y^{5}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. One factor of \(y^{2}+5 y-24\) is \(y-3\)

Factor using the formula for the sum or difference of two cubes. $$125 x^{3}-64 y^{3}$$

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{2}+4=(x+2)(x+2)\)

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

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$10 x^{3} y-14 x^{2} y^{2}+4 x y^{3}$$

Find all positive integers \(b\) so that the trinomial can be factored. \(x^{2}+b x+15\)

An explosion causes debris to rise vertically with an initial velocity of 64 feet per second. The polynomial \(64 x-16 x^{2}\) describes the height of the debris above the ground, in feet, after \(x\) seconds. a. Find the height of the debris after 3 seconds. b. Factor the polynomial. c. Use the factored form of the polynomial in part (b) to find the height of the debris after 3 seconds. Do you get the same answer as you did in part (a)? If so, does this prove that your factorization is correct? Explain.

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$25 x^{2}-\frac{4}{49}$$

Explain how to factor \(2 x^{2}-x-1\).

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$18 x^{3} y+57 x^{2} y^{2}+30 x y^{3}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Both 0 and \(-\pi\) are solutions of the equation \(x(x+\pi)=0\)

An explosion causes debris to rise vertically with an initial velocity of 72 feet per second. The polynomial \(72 x-16 x^{2}\) describes the height of the debris above the ground, in feet, after \(x\) seconds. a. Find the height of the debris after 4 seconds. b. Factor the polynomial. c. Use the factored form of the polynomial in part (b) to find the height of the debris after 4 seconds. Do you get the same answer as you did in part (a)? If so, does this prove that your factorization is correct? Explain

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$16 x^{2}-\frac{9}{25}$$

Why is it a good idea to factor out the GCF first and then use other methods of factoring? Use \(3 x^{2}-18 x+15\) as an example. Discuss what happens if one first uses trial and error to factor as two binomials rather than first factoring out the GCF.