Polynomials and Factoring

Polynomials and FactoringWhat is a polynomial?Any finite sum of terms.

6x + 2x 15x 3 15

trinomial of degree 2 binomial of degree 1 polynomial of degree 6 monomial of degree 4 monomial of degree 1Adding & SubtractingTo add or subtract two polynomials, add or subtract the like terms.

Like termsAny two terms with the same variables and exponents

Adding & SubtractingWhich of the following are like terms?

1) 7xy and 6x

2) 3y and -4y

3) 5x and 3x

4) 4xy and -2xy not like terms (variables are different) These are like terms not like terms (exponents are different) not like terms (exponents)Adding & SubtractingFind the sum:

(5x - x + 2x + 7) + (3x + 7 4x) + (4x - 8 - x)

Starting with the highest degree, what are like terms?= 4x+ 9x- 5x+ 6Adding & SubtractingFind the sum:

(2x + x - 5) + (x + x + 6)= 3x + 2x+ 1Adding & SubtractingAdd the following polynomials:(12x + 10) + (18x - 3x + 6)

(4x - 7x + 2) + (-x + x 2)

(8y + 4y + 3y 7) + (2y - 6y + 4)

(x - 6x) + (2x + 9) + (4x + x)(12x + 10) + (18x - 3x + 6)= 30x- 3x+ 16(4x - 7x + 2) + (-x + x 2)= 3x- 6x(8y + 4y + 3y 7) + (2y - 6y + 4)= 8y+ 6y- 3y- 3(x - 6x) + (2x + 9) + (4x + x)= 4x+ 4x- 6x+ 9Polynomials & FactoringAdding & SubtractingFind the sum:

(2x - 3x + 6x + 7) + (-3x - 7 + x) + (2x - x)

Starting with the highest degree, what are like terms?= x+ 5x- 2xSubtractingSo far, the only operation we have used has been addition.

Most of the mistakes made on MCAS tests are on Subtraction problemsSubtractionFind the difference

(5x + 2x 4) (3x - 3x + 6)

You must distribute the negative sign over the polynomial in the immediately parentheses to the right

5x + 2x 4-3x+ 6x-6= 2x+ 8x- 10SubtractingFind the difference between the polynomials.

(5h + 4h + 8) (3h + h + 3)

Re-write the equation after distributing the minus sign

5h + 4h + 8 + -3h- h -3= 2h+ 3h+ 5Subtracting(-7z + 2z 7) (2z - z 3)

-7z + 2z - 7+ -2z+ z+ 3= -9z+ 3z- 4PerimeterThe perimeter of a polygon is the sum of the lengths of its sides. Write a simplified expression for the perimeter of this polygon.x + 24x 23x + 152x + 3+ x + 2+ 4x 2+ 3x + 152x + 3Per. = Per. = 10x+ 18PerimeterThe perimeter of a polygon is the sum of the lengths of its sides. Write a simplified expression for the perimeter of this polygon.x + 4x 3x - 1x - 3+ x + 4+ x 3+ x - 1x - 3Per. = Per. = 4x- 3Polynomials & FactoringSo far we have:Added polynomials(4x + 2x + 3) + (2x - x - 2)

Subtracted polynomials(3x + 4x + 8) (2x + 3x 5) 3x + 4x + 86x+ x+ 1- 2x- 3x+ 5x+ x+ 13Multiplying PolynomialsWhen we multiply two binomials

(x + 2) (2x 4)

Use the F.O.I.L. method

F - First O - OuterI - InnerL - Last F.O.I.L(x + 2) (2x 4) =F - Firstx 2x2xO - Outerx - 4- 4xI - Inner2 2x+ 4xL - Last2 -4- 8= 2x - 8F.O.I.L (x + 2) (x 3)x x= x= xx -3= -3x+ -3x2 x= 2x+ 2x2 -3= -6+ -6= x+ -1x- 6Multiply the following binomials.(2x 3) (x + 4)

(3x + 1) (x + 5)

(x 4) (x + 7)

(4x + 4) (2x 3)F.O.I.L (2x - 3) (x + 4)2x x= 2x= 2x2x 4= 8x+ 8x-3 x= -3x+ -3x-3 4= -12+ -12= 2x+ 5x- 12F.O.I.L (3x + 1) (x + 5)3x x= 3x= 3x3x 5= 15x+ 15x1 x= x+ x1 5= 5+ 5= 3x+ 16x+ 5F.O.I.L (x - 4) (x + 7)x x= x= xx 7= 7x+ 7x-4 x= -4x+ -4x-4 7= -28- 28= x+ 3x- 28F.O.I.L (4x + 4) (2x - 3)4x 2x= 8x= 8x4x -3= -12x+ -12x4 2x= 8x+ 8x4 -3= -12 - 12= 8x - 4x- 12Polynomials & FactoringThis week:Added polynomials

Subtracted polynomials

Multiplied binomials (F.O.I.L.)Adding Polynomials(x + 3x 4) + (3x + 5x 8)(2x 7) + (4x + 3x + 3)= 4x+ 8x- 12= 6x+ 3x- 4Subtracting Polynomials(5x + 2x 7) - (2x + 4x 3)= 5x + 2x - 7- 2x+ 3- 4x= 3x- 4- 2xF.O.I.L (2x + 3) (4x - 1)2x 4x= 8x= 8x2x -1= -2x+ -2x3 4x= 12x+ 12x3 -1= -3 - 3= 8x + 10x- 3