Polynomials

Let’s Define it.

POLYNOMIALS: A REVIEW

P O L Y N O M I A L V O C A B U L A R Y

Term – a number or a product of a number and variables raised to powers

Coefficient – numerical factor of a term

Constant – term which is only a number

Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

P O L Y N O M I A L V O C A B U L A R Y

In the polynomial 7x5 + x2y2 – 4xy + 7

There are 4 terms: 7x5, x2y2, -4xy and 7.

The coefficient of term 7x5 is 7,

of term x2y2 is 1,

of term –4xy is –4 and

of term 7 is 7.

7 is a constant term.

T Y P E S O F P O L Y N O M I A L S

Monomial is a polynomial with 1 term.

Binomial is a polynomial with 2 terms.

Trinomial is a polynomial with 3 terms.

Multinomial is a polynomial with 4 or more terms.

D E G R E E S

Degree of a term

To find the degree, take the sum of the exponents on the variables contained in the term.

Degree of a constant is 0.

Degree of the term 5a4b3c is 8 (remember that c can be written as c1).

Degree of a polynomial

To find the degree, take the largest degree of any term of the polynomial.

Degree of 9x3 – 4x2 + 7 is 3.

E V A L U A T I N G P O L Y N O M I A L S

Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved.

Find the value of 2x3 – 3x + 4 when x = 2.

= 2( 2)3 – 3( 2) + 42x3 – 3x + 4

= 2( 8) + 6 + 4

= 6

Example

C O M B I N I N G L I K E T E R M S

Like terms are terms that contain exactly the same variables raised to exactly the same powers.

Combine like terms to simplify.

x2y + xy – y + 10x2y – 2y + xy

Only like terms can be combined through addition and subtraction.

Warning!

11x2y + 2xy – 3y= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y =

= x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together)

Example

Let’s Add and Subtract!

ADDING AND SUBTRACTING POLYNOMIALS


Adding Polynomials

Combine all the like terms.

Subtracting Polynomials

Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms.


= 3a2 – 6a + 11

Add or subtract each of the following, as indicated.

1) (3x – 8) + (4x2 – 3x +3)

= 4x2 + 3x – 3x – 8 + 3

= 4x2 – 5

2) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8

3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)

= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7

= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7

= 3x – 8 + 4x2 – 3x + 3

Example

In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression.

You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically.


Let’s Multiply!

MULTIPLYING POLYNOMIALS

M U L T I P L Y I N G P O L Y N O M I A L S

Multiplying polynomials

• If all of the polynomials are monomials, use the associative and commutative properties.

• If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms.

Multiply each of the following.

1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3

2) (4x2)(3x2 – 2x + 5)

= (4x2)(3x2) – (4x2)(2x) + (4x2)(5) (Distributive property)

= 12x4 – 8x3 + 20x2 (Multiply the monomials)

3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5)= 14x2 + 10x – 28x – 20= 14x2 – 18x – 20

M u l t i p l y i n g P o l y n o m i a l s

Example

Multiply (3x + 4)2

Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4).

(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4)

= 9x2 + 12x + 12x + 16

= 9x2 + 24x + 16

Multiplying Polynomials

Example

Multiply (a + 2)(a3 – 3a2 + 7).

(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7)

= a4 – 3a3 + 7a + 2a3 – 6a2 + 14

= a4 – a3 – 6a2 + 7a + 14

Multiplying PolynomialsExample

Multiply (3x – 7y)(7x + 2y)

(3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y)

= 21x2 + 6xy – 49xy + 14y2

= 21x2 – 43xy + 14y2


Multiply (5x – 2z)2

(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z)

= 25x2 – 10xz – 10xz + 4z2

= 25x2 – 20xz + 4z2


Multiply (2x2 + x – 1)(x2 + 3x + 4)

(2x2 + x – 1)(x2 + 3x + 4)

= (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4)

= 2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4

= 2x4 + 7x3 + 10x2 + x – 4

Multiplying Polynomials

Example

Let’s multiply!

SPECIAL PRODUCTS

THE FOIL METHOD

When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method.

F – product of First terms

O – product of Outside terms

I – product of Inside terms

L – product of Last terms

= y2 – 8y – 48

Multiply (y – 12)(y + 4)

(y – 12)(y + 4)

(y – 12)(y + 4)

(y – 12)(y + 4)

(y – 12)(y + 4)

Product of First terms is y2

Product of Outside terms is 4y

Product of Inside terms is -12y

Product of Last terms is -48

(y – 12)(y + 4) = y2 + 4y – 12y – 48F O I L

Using the FOIL Method

Example

Multiply (2x – 4)(7x + 5)

(2x – 4)(7x + 5) =

= 14x2 + 10x – 28x – 20

F

2x(7x)F

+ 2x(5)O

– 4(7x)I

– 4(5)L

O

I

L

= 14x2 – 18x – 20We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product.

Example

Using the FOIL Method

In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products.

Squaring a Binomial

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

Multiplying the Sum and Difference of Two Terms

(a + b)(a – b) = a2 – b2

Special Products

Although you will arrive at the same results for the special products by using the techniques of this section or last section, memorizing these products can save you some time in multiplying polynomials.

Special Products

DIVIDING POLYNOMIALS

Let’s divide!

D I V I D I N G P O L Y N O M I A L S

Dividing a polynomial by a monomial

Divide each term of the polynomial separately by the monomial.

a

aa

3

153612 3 aa

a

a

a

3

15

3

36

3

12 3

aa

5124 2

Example

DIVIDING POLYNOMIALS

Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide.

D I V I D I N G P O L Y N O M I A L S

725643 7256431

4329

6

2585

37

8

6344

32

Divide 43 into 72.

Multiply 1 times 43.

Subtract 43 from 72.

Bring down 5.

Divide 43 into 295.



Bring down 6.

Divide 43 into 376.



Nothing to bring down.32168 .43

We then write our result as

DIVIDING POLYNOMIALSD I V I D I N G P O L Y N O M I A L S

As you can see from the previous example, there is a pattern in the long division technique.

DivideMultiplySubtractBring downThen repeat these steps until you can’t bring down or divide any longer.

We will incorporate this same repeated technique with dividing polynomials.

Dividing PolynomialsD I V I D I N G P O L Y N O M I A L S

15232837 2 xxx

x4

xx 1228 2 35- x

5

1535 x

Divide 7x into 28x2.

Multiply 4x times 7x+3.

Subtract 28x2 + 12x from 28x2 – 23x.

Bring down – 15.

Divide 7x into –35x.

Multiply – 5 times 7x+3.

Subtract –35x–15 from –35x–15.

Nothing to bring down.

15-

So our answer is 4x – 5.

DIVIDING POLYNOMIALSD I V I D I N G P O L Y N O M I A L S

86472 2 +-+ xxx

x2

xx 144 2+20- x

10-

7020 -- x78

Divide 2x into 4x2.

Multiply 2x times 2x+7.

Subtract 4x2 + 14x from 4x2 – 6x.

Bring down 8.

Divide 2x into –20x.

Multiply -10 times 2x+7.

Subtract –20x–70 from –20x+8.

Nothing to bring down.

8+

+)72(

78+x

x2 10-We write our final answer as

Dividing PolynomialsD I V I D I N G P O L Y N O M I A L S

THE END

GOODBYE!

Polynomials

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