Chapter 3

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(2 x^{n}\right)\left(-5 x^{n}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$18 x^{2}-12 x+2$$

Solve each equation for the indicated variable. \(3 a y^{2}=b y\) for \(y\)

Find the indicated products. Assume all variables that appear as exponents represent positive integers. $$\left(3 x^{n}-7\right)^{2}$$

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(4 x^{2 n-1}\right)\left(-3 x^{n+1}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$x^{4}-5 x^{2}-36$$

Solve each equation for the indicated variable. \(y^{2}-a y+2 b y-2 a b=0\) for \(y\)

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(-3 a^{2}\right)\left(-4 a^{n+2}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$6 x^{4}-5 x^{2}-21$$

Solve each equation for the indicated variable. \(x^{2}+a x+b x+a b=0\) for \(x\)

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(-5 x^{n-1}\right)\left(-6 x^{2 n+4}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$6 w^{2}-11 w-35$$

Set up an equation and solve each of the following problems. The square of a number equals seven times the number. Find the number.

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(x^{n}\right)\left(2 x^{2 n}\right)\left(3 x^{2}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$10 x^{3}+15 x^{2}+20 x$$

Set up an equation and solve each of the following problems. Suppose that the area of a square is six times its perimeter. Find the length of a side of the square.

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(2 x^{n}\right)\left(3 x^{3 n-1}\right)\left(-4 x^{2 n+5}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 n^{2}+64$$

Set up an equation and solve each of the following problems. The area of a circular region is numerically equal to three times the circumference of the circle. Find the length of a radius of the circle.

A square piece of cardboard is 16 inches on a side. A square piece \(x\) inches on a side is cut out from each corner. The flaps are then turned up to form an open box. Find polynomials that represent the volume and outside surface area of the box.

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(3 x^{n-1}\right)\left(x^{n+1}\right)\left(4 x^{2-n}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$4 x^{2}-37 x+40$$

Set up an equation and solve each of the following problems. Find the length of a radius of a circle such that the circumference of the circle is numerically equal to the area of the circle.

How would you simplify \(\left(2^{3}+2^{2}\right)^{2} ?\) Explain your reasoning.

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(-5 x^{n+2}\right)\left(x^{n-2}\right)\left(4 x^{3-2 n}\right)$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$2 n^{3}+14 n^{2}-20 n$$

Set up an equation and solve each of the following problems. Suppose that the area of a circle is numerically equal to the perimeter of a square and that the length of a radius of the circle is equal to the length of a side of the square. Find the length of a side of the square. Express your answer in terms of \(\pi\).

Describe the process of multiplying two polynomials.

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 t^{2}-100$$

Set up an equation and solve each of the following problems. Find the length of a radius of a sphere such that the surface area of the sphere is numerically equal to the volume of the sphere.

Determine the number of terms in the product of \((x+y)\) and \((a+b+c+d)\) without doing the multiplication. Explain how you arrived at your answer.

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$2 x y+6 x+y+3$$

Set up an equation and solve each of the following problems. Suppose that the area of a square lot is twice the area of an adjoining rectangular plot of ground. If the rectangular plot is 50 feet wide, and its length is the same as the length of a side of the square lot, find the dimensions of both the square and the rectangle.

We have used the following two multiplication patterns. $$ \begin{aligned} &(a+b)^{2}=a^{2}+2 a b+b^{2} \\ &(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \end{aligned} $$ By multiplying, we can extend these patterns as follows: $$ \begin{aligned} &(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ &(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a^{4}+b^{5} \end{aligned} $$ On the basis of these results, see if you can determine a pattern that will enable you to complete each of the following without using the long- multiplication process. (a) \((a+b)^{6}\) (b) \((a+b)^{7}\) (c) \((a+b)^{8}\) (d) \((a+b)^{9}\)

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$3 x y+15 x-2 y-10$$

Set up an equation and solve each of the following problems. The area of a square is one-fourth as large as the area of a triangle. One side of the triangle is 16 inches long, and the altitude to that side is the same length as a side of the square. Find the length of a side of the square.

Find each of the following indicated products. These patterns will be used again in Section 3.5. (a) \((x-1)\left(x^{2}+x+1\right)\) (b) \((x+1)\left(x^{2}-x+1\right)\) (c) \((x+3)\left(x^{2}-3 x+9\right)\) (d) \((x-4)\left(x^{2}+4 x+16\right)\) (e) \((2 x-3)\left(4 x^{2}+6 x+9\right)\) (f) \((3 x+5)\left(9 x^{2}-15 x+25\right)\)

How would you convince someone that \(x^{6} \div x^{2}\) is \(x^{4}\) and not \(x^{3} ?\)

How can you determine that \(x^{2}+5 x+12\) is not factorable using integers?

Set up an equation and solve each of the following problems. Suppose that the volume of a sphere is numerically equal to twice the surface area of the sphere. Find the length of a radius of the sphere.

Some of the product patterns can be used to do arithmetic computations mentally. For example, let's use the pattern \((a+b)^{2}=a^{2}+2 a b+b^{2}\) to compute \(31^{2}\) mentally. Your thought process should be " \(31^{2}=\) \((30+1)^{2}=30^{2}+2(30)(1)+1^{2}=961 . "\) Compute each of the following numbers mentally, and then check your answers. (a) \(21^{2}\) (b) \(41^{2}\) (c) \(71^{2}\) (d) \(32^{2}\) (e) \(52^{2}\) (f) \(82^{2}\)

Your friend simplifies \(2^{3} \cdot 2^{2}\) as follows: $$ 2^{3} \cdot 2^{2}=4^{3+2}=4^{5}=1024 $$ What has she done incorrectly and how would you help her?

Explain your thought process when factoring \(30 x^{2}+13 x-56\)

Set up an equation and solve each of the following problems. Suppose that a radius of a sphere is equal in length to a radius of a circle. If the volume of the sphere is numerically equal to four times the area of the circle, find the length of a radius for both the sphere and the circle.

Use the pattern \((a-b)^{2}=a^{2}-2 a b+b^{2}\) to compute each of the following numbers mentally, and then check your answers. (a) \(19^{2}\) (b) \(29^{2}\) (c) \(49^{2}\) (d) \(79^{2}\) (e) \(38^{2}\) (f) \(58^{2}\)

Consider the following approach to factoring \(12 x^{2}+54 x+60\) $$ \begin{aligned} 12 x^{2}+54 x+60 &=(3 x+6)(4 x+10) \\ &=3(x+2)(2)(2 x+5) \\ &=6(x+2)(2 x+5) \end{aligned} $$ Is this a correct factoring process? Do you have any suggestion for the person using this approach?

Is \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11+7\) a prime or a composite number? Defend your answer.

Every whole number with a units digit of 5 can be represented by the expression \(10 x+5\), where \(x\) is a whole number. For example, \(35=10(3)+5\) and \(145=\) \(10(14)+5\). Now let's observe the following pattern when squaring such a number. $$ \begin{aligned} (10 x+5)^{2} &=100 x^{2}+100 x+25 \\ &=100 x(x+1)+25 \end{aligned} $$ The pattern inside the dashed box can be stated as "add 25 to the product of \(x, x+1\), and \(100 . "\) Thus, to compute \(35^{2}\) mentally, we can think " \(35^{2}=3(4)(100)+25\) =1225." Compute each of the following numbers mentally, and then check your answers. (a) \(15^{2}\) (b) \(25^{2}\) (c) \(45^{2}\) (d) \(55^{2}\) (e) \(65^{2}\) (f) \(75^{2}\) (g) \(85^{2}\) (h) \(95^{2}\) (i) \(105^{2}\)

Factor each trinomial and assume that all variables that appear as exponents represent positive integers. $$x^{2 a}+2 x^{a}-24$$

Suppose that your friend factors \(36 x^{2} y+48 x y^{2}\) as follows: $$ \begin{aligned} 36 x^{2} y+48 x y^{2} &=(4 x y)(9 x+12 y) \\ &=(4 x y)(3)(3 x+4 y) \\ &=12 x y(3 x+4 y) \end{aligned} $$ Is this a correct approach? Would you have any suggestion to offer your friend?