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# Chapter 9: Polynomials and Factoring

Difficulty Level: Basic Created by: CK-12

## Introduction

This chapter will present a new type of function: the polynomial. Chances are, polynomials will be new to you. However, polynomials are used in many careers and real life situations - to model the population of a city over a century, to predict the price of gasoline, and to predict the volume of a solid. This chapter will also present basic factoring - breaking a polynomial into its linear factors. This will help you solve many quadratic equations found in Chapter 10.

## Summary

This chapter first deals with polynomials. It starts by talking about how to classify, simplify, and rewrite polynomials in standard form, and it then moves on to discuss the adding and subtracting of polynomials. Next, instruction is given on multiplying a polynomial by a monomial, multiplying a polynomial by a binomial, and finding special products of polynomials. The chapter then provides a great deal of information on factors and factoring, including monomial factors of polynomials, factorization of quadratic expressions, using special products to factor, factoring by grouping, and factoring completely. Solving polynomial equations using the Zero Product Principle is also touched upon. Finally, the probability of compound events is introduced.

### Polynomials and Factoring; More on Probability Review

Define the following words:

1. Polynomial
2. Monomial
3. Trinomial
4. Binomial
5. Coefficient
6. Independent events
7. Factors
8. Factoring
9. Greatest common factor
10. Constant
11. Mutually exclusive
12. Dependent events

Identify the coefficients, constants, and the polynomial degrees in each of the following polynomials.

1. x53x3+4x25x+7\begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}
2. x43x3y2+8x12\begin{align*}x^4-3x^3y^2+8x-12\end{align*}

Rewrite the following in standard form.

1. 4b+4+b2\begin{align*}-4b+4+b^2\end{align*}
2. 3x2+5x42x+9\begin{align*}3x^2+5x^4-2x+9\end{align*}

Add or subtract the following polynomials and simplify.

1. Add x22xy+y2\begin{align*}x^2-2xy+y^2\end{align*} and 2y24x2\begin{align*}2y^2-4x^2\end{align*} and 10xy+y3\begin{align*}10xy+y^3\end{align*}.
2. Subtract x33x2+8x+12\begin{align*}x^3-3x^2+8x+12\end{align*} from 4x2+5x9\begin{align*}4x^2+5x-9\end{align*}.
3. Add 2x3+3x2y+2y\begin{align*}2x^3+3x^2 y+2y\end{align*} and x32x2y+3y\begin{align*}x^3-2x^2 y+3y\end{align*}

Multiply and simplify the following polynomials.

1. (3y4)(2y2)\begin{align*}(-3y^4)(2y^2)\end{align*}
2. 7a2bc3(5a23b29c2)\begin{align*}-7a^2bc^3(5a^2-3b^2-9c^2)\end{align*}
3. 7y(4y22y+1)\begin{align*}-7y(4y^2-2y+1)\end{align*}
4. (3x2+2x5)(2x3)\begin{align*}(3x^2+2x-5)(2x-3)\end{align*}
5. (x29)(4x4+5x22)\begin{align*}(x^2-9)(4x^4+5x^2-2)\end{align*}
6. (2x3+7)(2x37)\begin{align*}(2x^3+7)(2x^3-7)\end{align*}

Square the binomials and simplify.

1. (x2+4)2\begin{align*}(x^2+4)^2\end{align*}
2. (5x2y)2\begin{align*}(5x-2y)^2\end{align*}
3. (13x2+2y)\begin{align*}(13x^2+2y)\end{align*}

Solve the following polynomial equations.

1. 4x(x+6)(4x9)=0\begin{align*}4x(x+6)(4x-9)=0\end{align*}
2. x(5x4)=0\begin{align*}x(5x-4)=0\end{align*}

Factor out the greatest common factor of each expression

1. 12n+28n+4\begin{align*}-12n+28n+4\end{align*}
2. 45x10+45x7+18x4\begin{align*}45x^{10}+45x^7+18x^4\end{align*}
3. 16y58y5x2+40y6x3\begin{align*}-16y^5-8y^5 x^2+40y^6 x^3\end{align*}
4. 15u410u610u3v\begin{align*}15u^4-10u^6-10u^3 v\end{align*}
5. 6a9+20a4b+10a3\begin{align*}-6a^9+20a^4 b+10a^3\end{align*}
6. 12x+27y227x6\begin{align*}12x+27y^2-27x^6\end{align*}

Factor the difference of squares.

1. x2100\begin{align*}x^2-100\end{align*}
2. x21\begin{align*}x^2-1\end{align*}
3. 16x225\begin{align*}16x^2-25\end{align*}
4. 4x281\begin{align*}4x^2-81\end{align*}

Factor the following expressions completely.

1. 5n2+25n\begin{align*}5n^2+25n\end{align*}
2. 7r2+37r+36\begin{align*}7r^2+37r+36\end{align*}
3. 4v2+36v\begin{align*}4v^2+36v\end{align*}
4. 336xy288x2+294y252x\begin{align*}336xy-288x^2+294y-252x\end{align*}
5. 10xy25x+8y20\begin{align*}10xy-25x+8y-20\end{align*}

Complete the following problems.

1. One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the right triangle.
2. A rectangle has sides of x+5\begin{align*}x+5\end{align*} and x3\begin{align*}x-3\end{align*}. What value of x\begin{align*}x\end{align*} gives an area of 48?
3. Are these two events mutually exclusive, mutually inclusive, or neither? “Choosing the sports section from a newspaper” and “choosing the times list for a movie theater”
4. You spin a spinner with seven equal sections numbered one through seven and roll a six-sided cube. What is the probability that you roll a five on both the cube and the spinner?
5. You spin a spinner with seven equal sections numbered one through seven and roll a six-sided cube. Are these events mutually exclusive?
6. You spin a spinner with seven equal sections numbered one through seven and roll a six-sided cube. Are these events independent?
7. You spin a spinner with seven equal sections numbered one through seven and roll a six-sided cube. What is the probability you spin a 3, 4, or 5 on the spinner or roll a 2 on the cube?

### Polynomials and Factoring; More on Probability Test

Simplify the following expressions.

1. (4x2+5x+1)(2x2x3)\begin{align*}(4x^2+5x+1)-(2x^2-x-3)\end{align*}
2. (2x+5)(x2+3x4)\begin{align*}(2x+5)-(x^2+3x-4)\end{align*}
3. (b+4c)+(6b+2c+3d)\begin{align*}(b+4c)+(6b+2c+3d)\end{align*}
4. (5x2+3x+3)+(3x26x+4)\begin{align*}(5x^2+3x+3)+(3x^2-6x+4)\end{align*}
5. (3x+4)(x5)\begin{align*}(3x+4)(x-5)\end{align*}
6. (9x2+2)(x3)\begin{align*}(9x^2+2)(x-3)\end{align*}
7. (4x+3)(8x2+2x+7)\begin{align*}(4x+3)(8x^2+2x+7)\end{align*}

Factor the following expressions.

1. 27x218x+3\begin{align*}27x^2-18x+3\end{align*}
2. 9n2100\begin{align*}9n^2-100\end{align*}
3. 648x232\begin{align*}648x^2-32\end{align*}
4. 81p290p+25\begin{align*}81p^2-90p+25\end{align*}
5. 6x235x+49\begin{align*}6x^2-35x+49\end{align*}

Solve the following problems.

1. A rectangle has sides of x+7\begin{align*}x+7\end{align*} and x5\begin{align*}x-5\end{align*}. What value of x\begin{align*}x\end{align*} gives an area of 63?
2. The product of two positive numbers is 50. Find the two numbers if one of the numbers is 6 more than the other.
3. Give an example of two independent events. Determine the probability of each event. Use it to find:
1. P(AB)\begin{align*}P(A \cup B)\end{align*}
2. P(AB)\begin{align*}P(A \cap B)\end{align*}
1. The probability it will rain on any given day in Seattle is 45%. Find the probability that:
1. It will rain three days in a row.
2. It will rain one day, not the next, and rain again on the third day.

#### Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9619.

Basic

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Feb 24, 2012