Factorization of polynomials and factor theorem

Factorization of polynomials and factor theorem

Open Ended

What do you know about the factor theorem?

Factorization of polynomials

• Let 2x+2 be p(x)

• Divide 2x+2 by x+1

• You'll find that the remainder is zero.

• Note: If we divide a polynomial p(x) by another polynomial g(x) and r(x)=0, then g(x) is a zero of the polynomial p(x).

Video Time

https://www.youtube.com/watch?v=H9MHo5UM7oI

Factor theorem

Factor Theorem : If p(x) is a polynomial of degree n > 1 and a is any real number,

then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).

Proof of the factor theorem

Proof: By the Remainder Theorem, p(x)=(x – a) q(x) + p(a).

(i) If p(a) = 0, then p(x) = (x – a) q(x), which shows that x – a is a factor of p(x).

(ii) Since x – a is a factor of p(x), p(x) = (x – a) g(x) for same polynomial g(x).

In this case, p(a) = (a – a) g(a) = 0.

Questions

• Examine whether x + 2 is a factor of x3 + 3x2 + 5x + 6 and of 2x + 4.

• Example 12 : Find the value of k, if x – 1 is a factor of 4x3 + 3x2 – 4x + k

Splitting the middle term

https://www.youtube.com/watch?v=YMBjk_v-0j8

Questions

• Factorise 6x2 + 17x + 5 by splitting the middle term.

• Factorise y2 – 5y + 6 by using splitting the middle term.

• Factorise x3 – 23x2 + 142x – 120.

Poll

Any doubts?

• Yes

• No

Multiple Choice

Using factor theorem, find whether 4x+8 is a multiple of x+2

• Yes

• No

• Don't know

Open Ended

What did you understand from the remainder theorem?

Multiple Select

How can we split

 $2x^2+7x+6$  ?

•  $2x^2+3x+4x+6$  

•  $2x+3x+4x+6$  

•  $2x^2+3x\times-4x+6$  

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