Question

Divide and check whether the g(x) is factor of p(x) by division

algorithm.

(a) p(x)=x4-3x3+4x2-8x+4, g(x) =x2-x+2

(b) p(x) = 2x4-3x3-3x2+6x-2, g(x)=x2-2​

Answer 1

Step-by-step explanation:

x power 4 - 3xcube+4xsquare-8x+2,g(x)=xsquare -x+2

Answer 2

Concept

We can check whether the polynomial is a factor of another polynomial by long division method or synthetic division and if the remainder comes out to be 0 then the polynomial is a factor of other polynomial.  

Given

(a) p(x)=x⁴-3x³+4x²-8x+4, g(x) =x²-x+2

(b) p(x) = 2x⁴-3x³-3x²+6x-2, g(x)=x²-2​

Find

Check whether g(x) is factor of p(x) by division.

Solution

Part (a)

p(x)=x⁴-3x³+4x²-8x+4

g(x) =x²-x+2

Using long division method

Step 1

Divided the leading term of the dividend by the leading term of the divisor: i.e x⁴/x²=x².

Multiplied it by the divisor: x²(x²−x+2)=x⁴−x³+2x².

Subtracted the dividend from the obtained result: (x⁴−3x³+4x²−8x+4)−(x⁴−x³+2x²)=−2x³+2x²−8x+4.

Step 2

Divided the leading term of the obtained remainder by the leading term of the divisor: i.e −2x³/x²=−2x.

Multiplied it by the divisor: −2x(x²−x+2)=−2x³+2x²−4x.

Subtracted the remainder from the obtained result: (−2x³+2x²−8x+4)−(−2x³+2x²−4x)=−4x+4.

Since the degree of the remainder is less than the degree of the divisor which means that g(x) is not a factor of p(x).

Part (b)

p(x) = 2x⁴-3x³-3x²+6x-2

g(x)=x²-2​

Using long division method

Step 1

Divided the leading term of the dividend by the leading term of the divisor: 2x⁴/x²=2x².

Multiplied it by the divisor: 2x²(x²−2)=2x⁴−4x².

Subtracted the dividend from the obtained result: (2x⁴−3x³−3x²+6x−2)−(2x⁴−4x²)=−3x³+x²+6x−2.

Step 2

Divided the leading term of the obtained remainder by the leading term of the divisor: i.e  −3x³/x²=−3x.

Multiplied it by the divisor: −3x(x²−2)=−3x³+6x.

Subtracted the remainder from the obtained result: (−3x³+x²+6x−2)−(−3x³+6x)=x²−2.

Step 3

Divided the leading term of the obtained remainder by the leading term of the divisor: i.e  x²/x²=1.

Multiplied it by the divisor: 1(x²−2)=x²−2.

Subtracted the remainder from the obtained result: (x²−2)−(x²−2) =0.

Since the remainder is 0 which means g(x) is factor of p(x).

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