4 Quadratics and polynomials

Lesson

An expression of the form Ax^{n}, where A is any number and n is any non-negative integer, is called a **monomial**. When we take the sum of multiple monomials, we get a **polynomial**.

In the monomial Ax^{n}:

A is the

**coefficient**x is the

**variable**n is the

**index**

A polynomial is a sum of any number of monomials (and consequently, each term of a polynomial is a monomial). The highest index is called the **degree** of the polynomial. For example, x^{3}+4x+3 is a polynomial of degree three. The coefficient of the term with the highest index is called the **leading coefficient**. The coefficient of the term with index 0 is called the **constant**.

We often name polynomials using function notation. For example P(x) is a polynomial where x is the variable. If we write a constant instead of x, that means that we substitute that constant for the variable. For example, if P(x)=x^{3}+4x+3 then P(3)=3^{3}+4\times 3+3=42.

Polynomials of particular degrees are given specific names. Some of these we have seen before.

Degree | Name |
---|---|

Zero | Constant |

One | Linear |

Two | Quadratic |

Three | Cubic |

For the polynomial P(x)=\dfrac{x^7}{5}+\dfrac{x^6}{6}+5.

a

What's the degree of the polynomial?

Worked Solution

b

What's the leading coefficient of the polynomial?

Worked Solution

c

What's the constant term of the polynomial?

Worked Solution

Consider P(x)=4x^5+3x^{6}-8.

a

Find P(0).

Worked Solution

b

Find P(-4).

Worked Solution

Idea summary

A monomial is an expression of the form:

\displaystyle Ax^{n}

\bm{A}

is the **coefficient**

\bm{x}

is the **variable**

\bm{n}

is the **index**

A polynomial is a sum of any number of monomials. The highest index is called the **degree** of the polynomial. The coefficient of the term with the highest index is called the **leading coefficient**. The coefficient of the term with index 0 is called the **constant**.

We apply operations to polynomials in the same way as we apply operations to numbers. For addition and subtraction we add or subtract all of the terms in both polynomials and we simplify by collecting like terms. For multiplication we multiply each term in one polynomial by each term in the other polynomial similar to how we expand binomial products. Division is a more complicated case that we will look at in the next lesson .

A polynomial is a sum of any number of monomials. In a polynomial:

The highest index is the

**degree**The coefficient of the term with the highest index is the

**leading coefficient**The coefficient of the term with index 0 is the

**constant**

We apply operations to polynomials in the same way that we apply operations to numbers.

If P(x)=-5x^{2}-6x-6 and Q(x)=-7x+7, form a simplified expression for P(x)-Q(x).

Worked Solution

Simplify \left(3x^{3}-9x^{2}-8x-7\right)+\left(-7x^{3}-9x\right).

Worked Solution

Idea summary

A polynomial is a sum of any number of monomials. In a polynomial:

The highest index is the

**degree**The coefficient of the term with the highest index is the

**leading coefficient**The coefficient of the term with index 0 is the

**constant**

Investigate the concept of a polynomial and apply the factor and remainder theorems to solve problems.