SB 5
Divide p(x) by g(x) and write the quotient, the remainder, and the division equation for the following.
Practice Question 1
For each case, determine if g(x) is a factor of p(x).
Division by a Monomial
1.1
Division by a Binomial
p(x)
g(x)
g(x)
3x
x+1
9x
X-3
- 3x
x²
2x
(a) 9x2 – 3x + 12
(b) 3x5 - 15x3 + 6x
(C) x + 6x – 1
(d) 11x5 + 7x3 - 2x
(e) 8x3 - x2 + 2x - 1
(0) x2 + 5x + 6
(g) x6 - 4x4 + 3x2
(h) x3 + 6x - 1
(0) 3x5 - 15x3 + 6x
(0) x + 7x - 1
p(x)
(a) x3 - 2x2 - 4x - 1
(b) x3 - 3x2 + 4x + 50
(c) 4x3 - 12x2 + 14x - 3
(d) x3 - 6x2 + 2x - 4
(e) x² – 3x² + 5x - 3
(f) x4 - 3x2 + 4x + 5
(g) x4 - 5x +6
(h) 5x3 - 3x2 + 7x +5
(1) 4x² – 3x²+x-6
(1) 7x² - 4x²+6x +
2x-1
1-32
x² - 2
1-x
2-x²
x-7
2x-1
x²
5x2
-3x
x - 4
9x
-2x
Answers
Answer:
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ANSWER
A polynomial p(x) is defined as
⇒p(x)=g(x)q(x)+r(x)
where g(x)= divisor ; q(x)= quotient and r(x)= remainder
∴ p(x) can be found by multiplying g(x) with q(x) & adding r(x) to the product.
(i).g(x)=(x−2); q(x)=x
2
−x+1; r(x)=4
∴p(x)=(x−2)[x
2
−x+1]+4
=x
3
−x
2
+x−2x
2
+2x−2+4
=x
3
−3x
2
+3x+2
(ii).g(x)=(x+3); q(x)=2x
2
+x+5; r(x)=3x+1
∴p(x)=(x+3)[2x
2
+x+5]+(3x+1)
=2x
3
+x
2
+5x+6x
2
+3x+15+3x+1
=2x
3
+7x
2
+11x+16
(iii).g(x)=(2x+1); q(x)=x
3
+3x
2
−x+1; r(x)=0
∴p(x)=(2x+1)[x
3
+3x
2
−x+1]+(0)
=2x
4
+6x
3
−2x
2
+2x+x
3
+3x
2
−x+1
=2x
4
+7x
3
+x
2
+x+1
(iv).g(x)=(x−1); q(x)=x
3
−x
2
−x−1; r(x)=2x−4
∴p(x)=(x−1)[x
3
−x
2
−x−1]+(2x−4)
=x
4
−x
3
−x
2
−x−x
3
+x
2
+x+1+2x−4
=x
4
−2x
3
+2x−3
(v).g(x)=(x
2
+2x+1); q(x)=x
4
−2x
2
+5x−7; r(x)=4x+12
∴p(x)=(x
2
+2x+1)[x
4
−2x
2
+5x−7]+(4x+12)
=x
6
−2x
4
+5x
3
−7x
2
+2x
5
+4x
3
+10x
2
−14x+x
4
−2x
2
+5x−7+4x+12
=x
6
−x
4
+x
3
+x
2
+2x
5
−5x+5
=x
6
+2x
5
−x
4
+x
3
+x
2
−5x+5
Hence, solve.
The division of p(x)=x
4
−3x
2
−4 by g(x)=x+2 is as shown above:
Now we know that the division algorithm states that:
Dividend=(Divisor×quotient)+Remainder
Here, the dividend is x
4
−3x
2
−4, the divisor is x+2, the quotient is x
3
−2x
2
+x−2 and the remainder is 0, therefore,
x
4
−3x
2
−4=[(x+2)(x
3
−2x
2
+x−2)]+0
⇒x
4
−3x
2
−4=[x(x
3
−2x
2
+x−2)+2(x
3
−2x
2
+x−2)]+0
⇒x
4
−3x
2
−4=x
4
−2x
3
+x
2
−2x+2x
3
−4x
2
+2x−4+0
⇒x
4
−3x
2
−4=x
4
−2x
3
+2x
3
+x
2
−4x
2
−2x+2x−4
⇒x
4
−3x
2
−4=x
4
−3x
2
−4
Hence, the division algorithm is verified.
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