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## SAT

### Course: SAT > Unit 6

Lesson 4: Passport to Advanced Math: lessons by skill- Solving quadratic equations | Lesson
- Interpreting nonlinear expressions | Lesson
- Quadratic and exponential word problems | Lesson
- Manipulating quadratic and exponential expressions | Lesson
- Radicals and rational exponents | Lesson
- Radical and rational equations | Lesson
- Operations with rational expressions | Lesson
- Operations with polynomials | Lesson
- Polynomial factors and graphs | Lesson
- Graphing quadratic functions | Lesson
- Graphing exponential functions | Lesson
- Linear and quadratic systems | Lesson
- Structure in expressions | Lesson
- Isolating quantities | Lesson
- Function Notation | Lesson

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# Polynomial factors and graphs | Lesson

## What are polynomial factors and graphs, and how frequently do they appear on the test?

**Note:**On your official SAT, you'll likely see at most

**1 question**that tests your knowledge of polynomial functions and their graphs. Make sure you understand the more frequently-tested skills on the SAT before you spend practice time on this skill.

In a

**polynomial function**, the output of the function is based on a**polynomial expression**in which the input is raised to the second power or higher.**Quadratic functions**are a type of polynomial function. However, this lesson focuses on polynomial functions raised to the

**third power or higher**.

Consider the function f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6. This function is a

**third degree polynomial**; its highest exponent of x is 3, and it has three x-intercepts:In this lesson, we'll:

- Relate the factors of polynomial functions to the x-intercepts of polynomial graphs
- Apply the
**polynomial remainder theorem**

This lesson builds upon the following skills:

**You can learn anything. Let's do this!**

## How do I identify features of graphs from polynomial functions?

### Zeros of polynomials introduction

### Features of polynomial graphs

#### Factored form and zeros

**Note:**The terms "zeros", "roots", and "x-intercepts" are used interchangeably here and on the test!

On the SAT, polynomial functions are usually shown in

**factored form**. For example, the function above, f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6, would be written as f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis.This is because the factors tell us the

**x-intercepts**of the graph. At each x-intercept, the value of y is zero. By the**zero product property**, if any of the factors is equal to 0, then the entire polynomial expression is equal to 0. Therefore, for y, equals, left parenthesis, x, minus, start color #7854ab, a, end color #7854ab, right parenthesis, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, the x-intercepts of graph are located at left parenthesis, start color #7854ab, a, end color #7854ab, comma, 0, right parenthesis, left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis, and left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis.According to the graph of y, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis above, the graph intercepts the x-axis at start color #7854ab, minus, 3, end color #7854ab, start color #ca337c, minus, 1, end color #ca337c, and start color #208170, 2, end color #208170, which means their corresponding factors are equal to 0 when x is equal to start color #7854ab, minus, 3, end color #7854ab, start color #ca337c, minus, 1, end color #ca337c, and start color #208170, 2, end color #208170:

- start color #7854ab, minus, 3, end color #7854ab, plus, 3, equals, 0
- start color #ca337c, minus, 1, end color #ca337c, plus, 1, equals, 0
- start color #208170, 2, end color #208170, minus, 2, equals, 0

Higher order polynomials behave similarly. For any polynomial graph, the number of distinct x-intercepts is equal to the number of unique factors.

To determine the zeros of a polynomial function in factored form:

- Set each factor equal to 0.
- Solve the equations from Step 1. The solutions to the linear equations are the zeros of the polynomial function.

**Example:**What are the roots of y, equals, left parenthesis, 2, x, minus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 5, right parenthesis ?

To write a polynomial function when its zeros are provided:

- For each given zero, write a linear expression for which, when the zero is substituted into the expression, the value of the expression is 0.
- Each linear expression from Step 1 is a factor of the polynomial function.
- The polynomial function must include all of the factors without any additional unique binomial factors.

**Example:**The real roots of the polynomial function p, left parenthesis, x, right parenthesis are minus, 1, 3, and 8. Write a function that could be p, left parenthesis, x, right parenthesis.

#### Standard form, y-intercept, and end behavior

When a polynomial function is written in

**standard form**, e.g., y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6, we can't identify the zeros as easily, but we*can*determine the**y-intercept**and**end behavior**of the graph.The

**y-intercept**happens when x, equals, 0, and so is equal to the**constant term**of the polynomial expression. So, the y-intercept for y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6 is minus, 6.The highest power term tells us the end behavior of the graph. End behavior is just another term for what happens to the value of y as x becomes very large in both the positive and negative directions. For the highest power term start color #7854ab, a, end color #7854ab, x, start superscript, start color #ca337c, n, end color #ca337c, end superscript:

- If start color #7854ab, a, end color #7854ab, is greater than, 0, then y ultimately approaches positive infinity as x increases.
- If start color #7854ab, a, end color #7854ab, is less than, 0, then y ultimately approaches negative infinity as x increases.
- If start color #ca337c, n, end color #ca337c is
**even**, then the ends of the graph point in the same direction. - If start color #ca337c, n, end color #ca337c is
**odd**, then the ends of the graph point in different directions.

The highest power term in y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6 is start color #7854ab, 1, end color #7854ab, x, start superscript, start color #ca337c, 3, end color #ca337c, end superscript:

- Since start color #7854ab, 1, end color #7854ab, is greater than, 0, y approaches positive infinity as x increases.
- Since start color #ca337c, 3, end color #ca337c is odd, the other end of the graph points in the
*opposite*direction as x and x, cubed become more and more negative:*negative*infinity.

### Try it!

## What is the polynomial remainder theorem, and how do I apply it?

### Intro to the polynomial remainder theorem

### The polynomial remainder theorem

The

**polynomial remainder theorem**states that when a polynomial function p, left parenthesis, x, right parenthesis is divided by x, minus, a, the remainder of the division is equal to p, left parenthesis, a, right parenthesis.The polynomial remainder theorem lets us calculate the remainder without doing polynomial long division. It also tells us whether an expression x, minus, a is a factor of an unknown polynomial function as long as we know the value of p, left parenthesis, a, right parenthesis:

- If p, left parenthesis, a, right parenthesis, equals, 0, then left parenthesis, a, comma, 0, right parenthesis is an x-intercept, and x, minus, a is a factor of p, left parenthesis, x, right parenthesis.
- If p, left parenthesis, a, right parenthesis, does not equal, 0, then x, minus, a is
*not*a factor of p, left parenthesis, x, right parenthesis.

### Try it!

## Your turn!

## Things to remember

For a polynomial function in standard form, the constant term is equal to the y-intercept.

For the highest power term start color #7854ab, a, end color #7854ab, x, start superscript, start color #ca337c, n, end color #ca337c, end superscript in the standard form of a polynomial function:

- If start color #7854ab, a, end color #7854ab, is greater than, 0, then y ultimately approaches positive infinity as x increases.
- If start color #7854ab, a, end color #7854ab, is less than, 0, then y ultimately approaches negative infinity as x increases.
- If start color #ca337c, n, end color #ca337c is
**even**, then the ends of the graph point in the same direction. - If start color #ca337c, n, end color #ca337c is
**odd**, then the ends of the graph point in different directions.

The

**polynomial remainder theorem**states that when a polynomial function p, left parenthesis, x, right parenthesis is divided by x, minus, a, the remainder of the division is equal to p, left parenthesis, a, right parenthesis.## Want to join the conversation?

- Can someone please explain what exactly the remainder theorem is? I still don't fully understand how dividing a polynomial expression works.(8 votes)
- The polynomial remainder theorem states that if any given function f(x) is divided by a polynomial of the form (x - a), f(a) = the remainder of the polynomial division. Therefore, to calculate the remainder of any polynomial division, it is only necessary to substitute (a) for (x) in the original function. The remainder = f(a). If f(a) = 0, then a,0 is a zero of the function and (x-a) is a factor of the function. If f(a) is not = 0, then a is not a zero of the function and (x - a) is not a factor of the function.(14 votes)

- y ultimately approaches positive infinity as x increases. i dont understand what this means(2 votes)
- Think about the function's graph. That phrase deals with what would happen if you were to scroll to the right (positive x-direction) forever. If y approaches positive infinity as x increases, as you go to the right on the graph, the line goes upwards forever and doesn't stop. Examining what graphs do at their ends like this can be useful if you want to extrapolate some new information that you don't have data for.(5 votes)

- what is the polynomial remainder theorem?(0 votes)
- how to solve math(0 votes)