Question 1: Factorize each of the following algebraic expression:
x2 + 12x − 45
Answer 1: To factorise x2+12x−45, we will find two numbers p and q such that p+q=12 and pq=−45.
Now,
15+(−3)=12
and
15×(−3)=−45
Splitting the middle term 12x in the given quadratic as −3x+15x, we get:
x2+12x−45=x2−3x+15x−45
=(x2−3x)+(15x−45)
=x(x−3)+15(x−3)
=(x+15)(x−3)
Question 2: Factorize each of the following algebraic expression:
40 + 3x − x2
Answer 2: We have:
40+3x−x2
⇒−(x2−3x−40)
To factorise (x2−3x−40), we will find two numbers p and q such that p+q=−3 and pq=−40.
Now,
5+(−8)=−3
and 5×(−8)=−40 Splitting the middle term −3x in the given quadratic as 5x−8x, we get:
40+3x−x2=−(x2−3x−40)
=−(x2+5x−8x−40)
=−[(x2+5x)−(8x+40)]
=−[x(x+5)−8(x+5)]
=−(x−8)(x+5)
=(x+5)(−x+8)
Question 3: Factorize each of the following algebraic expression:
a2 + 3a − 88
Answer 3: To factorise a2+3a−88, we will find two numbers p and q such that p+q=3 and pq=−88.
Now, 11+(−8)=3 and 11×(−8)=−88Splitting the middle term 3a in the given quadratic as 11a−8a, we get:a2+3a−88=a2+11a−8a−88 =(a2+11a)−(8a+88)
=a(a+11)−8(a+11)
=(a−8)(a+11)
Question 4: Factorize each of the following algebraic expression:
a2 − 14a − 51
Answer 4: To factorise a2−14a−51, we will find two numbers p and q such that p+q=−14 and pq=−51.
Now,
3+(−17)=−14
and
3×(−17)=−51
Splitting the middle term −14a in the given quadratic as 3a−17a, we get:
a2−14a−51=a2+3a−17a−51
=(a2+3a)−(17a+51)
=a(a+3)−17(a+3)
=(a−17)(a+3)
Question 5: Factorize each of the following algebraic expression:
x2 + 14x + 45
Answer 5: To factorise x2+14x+45, we will find two numbers p and q such that p+q=14 and pq=45.
Now, 9+5=14 and 9×5=45Splitting the middle term 14x in the given quadratic as 9x+5x, we get:x2+14x+45=x2+9x+5x+45 =(x2+9x)+(5x+45)
=x(x+9)+5(x+9)
=(x+5)(x+9)
Question 6: Factorize each of the following algebraic expression:
x2 − 22x + 120
Answer 6: To factorise x2−22x+120, we will find two numbers p and q such that p+q=−22 and pq=120.
Now, (−12)+(−10)=−22 and (−12)×(−10)=120Splitting the middle term −22x in the given quadratic as −12x−10x, we get:x2−22x+12=x2−12x−10x+120 =(x2−12x)+(−10x+120)
=x(x−12)−10(x−12)
=(x−10)(x−12)
Question 7: Factorize each of the following algebraic expression:
x2 − 11x − 42
Answer 7: To factorise x2−11x−42, we will find two numbers p and q such that p+q=−11 and pq=−42.
Now,
3+(−14)=−22
and 3×(−14)=42Splitting the middle term −11x in the given quadratic as−14x+3x, we get: x2−11x−42=x2−14x+3x−42 =(x2−14x)+(3x−42)=x(x−14)+3(x−14)=(x+3)(x−14)Question 8: Factorize each of the following algebraic expression:
a2 + 2a − 3
Answer 8: To factorise a2+2a−3, we will find two numbers p and q such that p+q=2 and pq=−3.
Now, 3+(−1)=2 and 3×(−1)=−3Splitting the middle term 2a in the given quadratic as−a+3a, we get:a2+2a−3=a2−a+3a−3 =(a2−a)+(3a−3)=a(a−1)+3(a−1)=(a+3)(a−1)Question 9: Factorize each of the following algebraic expression:
a2 + 14a + 48
Answer 9: To factorise a2+14a+48, we will find two numbers p and q such that p+q=14 and pq=48.
Now, 8+6=14 and 8×6=48Splitting the middle term 14a in the given quadratic as 8a+6a, we get:a2+14a+48=a2+8a+6a+48 =(a2+8a)+(6a+48)=a(a+8)+6(a+8)=(a+6)(a+8)Question 10: Factorize each of the following algebraic expression:
x2 − 4x − 21
Answer 10: To factorise x2−4x−21, we will find two numbers p and q such that p+q=−4 and pq=−21.
Now,3+(−7)=−4 and 3×(−7)=−21Splitting the middle term −4x in the given quadratic as −7x+3x, we get:x2−4x−21=x2−7x+3x−21 =(x2−7x)+(3x−21)=x(x−7)+3(x−7)=(x+3)(x−7)Question 11: Factorize each of the following algebraic expression:
y2 + 5y − 36
Answer 11: To factorise y2+5y−36, we will find two numbers p and q such that p+q=5 and pq=−36.
Now,9+(−4)=5 and 9×(−4)=−36Splitting the middle term 5y in the given quadratic as −4y+9y, we get: y2+5y−36=y2−4y+9y−36 =(y2−4y)+(9y−36)=y(y−4)+9(y−4)=(y+9)(y−4)Question 12: Factorize each of the following algebraic expression:
(a2 − 5a)2 − 36
Answer 12: (a2−5a)2−36
=(a2−5a)2−62=[(a2−5a)−6][(a2−5a)+6]=(a2−5a−6)(a2−5a+6) In order to factorise a2−5a−6, we will find two numbers p and q such that p+q=−5 and pq=−6
Now, (−6)+1=−5 and (−6)×1=−6Splitting the middle term −5 in the given quadratic as −6a+a, we get:a2−5a−6=a2−6a+a−6 =(a2−6a)+(a−6) =a(a−6)+(a−6) =(a+1)(a−6) Now,
In order to factorise a2−5a+6, we will find two numbers p and q such that p+q=−5 and pq=6
Clearly,(−2)+(−3)=−5 and (−2)×(−3)=6Splitting the middle term −5 in the given quadrat