Dividing polynomials This PowerPoint presentation demonstrates two different methods of polynomial division. Click here to see algebraic long division Click here to see dividing “in your head”

Algebraic long division Divide 2x³ + 3x² - x + 1 by x + 2 x + 2 is the divisor The quotient will be here. 2x³ + 3x² - x + 1 is the dividend

Algebraic long division First divide the first term of the dividend, 2x³, by x (the first term of the divisor). This gives 2x². This will be the first term of the quotient.

Algebraic long division Now multiply 2x² by x + 2 and subtract

Algebraic long division Bring down the next term, -x.-x.

Algebraic long division Now divide –x², the first term of –x² - x, by x, the first term of the divisor which gives –x.–x.

Algebraic long division Multiply –x by x + 2 and subtract

Algebraic long division Bring down the next term, 1

Algebraic long division Divide x, the first term of x + 1, by x,x, the first term of the divisor which gives 1

Algebraic long division Multiply x + 2 by 1 and subtract

Algebraic long division The remainder is –1. The quotient is 2x² - x + 1

Dividing polynomials Click here to see this example of algebraic long division again Click here to see dividing “in your head” Click here to end the presentation

Dividing in your head Divide 2x³ + 3x² - x + 1 by x + 2 When a cubic is divided by a linear expression, the quotient is a quadratic and the remainder, if any, is a constant. Let the remainder be d. Let the quotient by ax² + bx + c 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d The first terms in each bracket give the term in x³x³ x multiplied by ax² gives ax³ so a must be 2.

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² + bx + c) + d The first terms in each bracket give the term in x³ x multiplied by ax² gives ax³ so a must be 2.

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² + bx + c) + d Now look for pairs of terms that multiply to give terms in x² x multiplied by bx gives bx² + 4x² must be 3x²3x² 2 multiplied by 2x² gives 4x²4x² so b must be

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² + -1x + c) + d Now look for pairs of terms that multiply to give terms in x² x multiplied by bx gives bx² bx² + 4x² must be 3x² 2 multiplied by 2x² gives 4x² so b must be -1.

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + c) + d Now look for pairs of terms that multiply to give terms in x x multiplied by c gives cx - 2x 2x must be -x-x 2 multiplied by -x -x gives -2x so c must be 1.

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) + d Now look for pairs of terms that multiply to give terms in x x multiplied by c gives cx cx - 2x must be -x 2 multiplied by -x gives -2x so c must be 1.

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) + d Now look at the constant term 2 multiplied by 1 gives d must be 1 then add d so d must be

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) - 1 Now look at the constant term 2 multiplied by 1 gives d must be 1 then add d so d must be -1.

Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) - 1 The quotient is 2x² - x + 1 and the remainder is –1.

Dividing polynomials Click here to see algebraic long division Click here to see this example of dividing “in your head” again Click here to end the presentation