Answer :
- Combine like terms where possible.
- Factor out the greatest common factor (GCF).
- Factor quadratic expressions into binomials, if possible.
- The factorised forms are:
a) $8x(2x + 7)$
b) $x(x+3)(x+14)$
c) $-3x^2(x^2 + 7)$
d) $x^2(x-5)(x-9)$
e) $2x(7x - 15)$
f) $x^2(x+4)(x-2)$
g) $x^3(x^3 - 18x + 72)$
h) $x^4(x - 91)$
### Explanation
1. Problem Analysis
We are asked to factorise the given polynomial expressions. We will identify common factors and use factoring techniques to simplify each expression.
2. Factorising a)
a) $x^2+15 x^2+56 x$
Combine like terms: $x^2 + 15x^2 = 16x^2$. So the expression becomes $16x^2 + 56x$.
Factor out the common factor $x$: $16x^2 + 56x = x(16x + 56)$.
Factor out 8 from the parenthesis: $x(16x + 56) = 8x(2x + 7)$.
3. Factorising b)
b) $x^3+17 x^2+42 x$
Factor out the common factor $x$: $x^3 + 17x^2 + 42x = x(x^2 + 17x + 42)$.
Factor the quadratic expression: $x(x^2 + 17x + 42) = x(x+3)(x+14)$.
4. Factorising c)
c) $x^4-4 x^4-21 x^2$
Combine like terms: $x^4 - 4x^4 = -3x^4$. So the expression becomes $-3x^4 - 21x^2$.
Factor out the common factor $x^2$: $-3x^4 - 21x^2 = -3x^2(x^2 + 7)$.
5. Factorising d)
d) $x^4-14 x^3+45 x^2$
Factor out the common factor $x^2$: $x^4 - 14x^3 + 45x^2 = x^2(x^2 - 14x + 45)$.
Factor the quadratic expression: $x^2(x^2 - 14x + 45) = x^2(x-5)(x-9)$.
6. Factorising e)
e) $x^2+13 x^2-30 x$
Combine like terms: $x^2 + 13x^2 = 14x^2$. So the expression becomes $14x^2 - 30x$.
Factor out the common factor $x$: $14x^2 - 30x = x(14x - 30)$.
Factor out 2 from the parenthesis: $x(14x - 30) = 2x(7x - 15)$.
7. Factorising f)
f) $x^4+2 x^3-8 x^2$
Factor out the common factor $x^2$: $x^4 + 2x^3 - 8x^2 = x^2(x^2 + 2x - 8)$.
Factor the quadratic expression: $x^2(x^2 + 2x - 8) = x^2(x+4)(x-2)$.
8. Factorising g)
g) $x^6-18 x^4+72 x^3$
Factor out the common factor $x^3$: $x^6 - 18x^4 + 72x^3 = x^3(x^3 - 18x + 72)$.
The cubic expression $x^3 - 18x + 72$ does not have obvious integer roots. We can leave it as is or try to find roots numerically.
9. Factorising h)
h) $x^5+11 x^4-102 x^4$
Combine like terms: $11x^4 - 102x^4 = -91x^4$. So the expression becomes $x^5 - 91x^4$.
Factor out the common factor $x^4$: $x^5 - 91x^4 = x^4(x - 91)$.
10. Final Answer
The factorised forms of the given expressions are:
a) $8x(2x + 7)$
b) $x(x+3)(x+14)$
c) $-3x^2(x^2 + 7)$
d) $x^2(x-5)(x-9)$
e) $2x(7x - 15)$
f) $x^2(x+4)(x-2)$
g) $x^3(x^3 - 18x + 72)$
h) $x^4(x - 91)$
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and science. For example, in physics, you might use factoring to solve equations that describe the motion of an object. In engineering, factoring can help simplify complex expressions when designing structures or circuits. In computer science, factoring is used in cryptography to break codes and secure data.
- Factor out the greatest common factor (GCF).
- Factor quadratic expressions into binomials, if possible.
- The factorised forms are:
a) $8x(2x + 7)$
b) $x(x+3)(x+14)$
c) $-3x^2(x^2 + 7)$
d) $x^2(x-5)(x-9)$
e) $2x(7x - 15)$
f) $x^2(x+4)(x-2)$
g) $x^3(x^3 - 18x + 72)$
h) $x^4(x - 91)$
### Explanation
1. Problem Analysis
We are asked to factorise the given polynomial expressions. We will identify common factors and use factoring techniques to simplify each expression.
2. Factorising a)
a) $x^2+15 x^2+56 x$
Combine like terms: $x^2 + 15x^2 = 16x^2$. So the expression becomes $16x^2 + 56x$.
Factor out the common factor $x$: $16x^2 + 56x = x(16x + 56)$.
Factor out 8 from the parenthesis: $x(16x + 56) = 8x(2x + 7)$.
3. Factorising b)
b) $x^3+17 x^2+42 x$
Factor out the common factor $x$: $x^3 + 17x^2 + 42x = x(x^2 + 17x + 42)$.
Factor the quadratic expression: $x(x^2 + 17x + 42) = x(x+3)(x+14)$.
4. Factorising c)
c) $x^4-4 x^4-21 x^2$
Combine like terms: $x^4 - 4x^4 = -3x^4$. So the expression becomes $-3x^4 - 21x^2$.
Factor out the common factor $x^2$: $-3x^4 - 21x^2 = -3x^2(x^2 + 7)$.
5. Factorising d)
d) $x^4-14 x^3+45 x^2$
Factor out the common factor $x^2$: $x^4 - 14x^3 + 45x^2 = x^2(x^2 - 14x + 45)$.
Factor the quadratic expression: $x^2(x^2 - 14x + 45) = x^2(x-5)(x-9)$.
6. Factorising e)
e) $x^2+13 x^2-30 x$
Combine like terms: $x^2 + 13x^2 = 14x^2$. So the expression becomes $14x^2 - 30x$.
Factor out the common factor $x$: $14x^2 - 30x = x(14x - 30)$.
Factor out 2 from the parenthesis: $x(14x - 30) = 2x(7x - 15)$.
7. Factorising f)
f) $x^4+2 x^3-8 x^2$
Factor out the common factor $x^2$: $x^4 + 2x^3 - 8x^2 = x^2(x^2 + 2x - 8)$.
Factor the quadratic expression: $x^2(x^2 + 2x - 8) = x^2(x+4)(x-2)$.
8. Factorising g)
g) $x^6-18 x^4+72 x^3$
Factor out the common factor $x^3$: $x^6 - 18x^4 + 72x^3 = x^3(x^3 - 18x + 72)$.
The cubic expression $x^3 - 18x + 72$ does not have obvious integer roots. We can leave it as is or try to find roots numerically.
9. Factorising h)
h) $x^5+11 x^4-102 x^4$
Combine like terms: $11x^4 - 102x^4 = -91x^4$. So the expression becomes $x^5 - 91x^4$.
Factor out the common factor $x^4$: $x^5 - 91x^4 = x^4(x - 91)$.
10. Final Answer
The factorised forms of the given expressions are:
a) $8x(2x + 7)$
b) $x(x+3)(x+14)$
c) $-3x^2(x^2 + 7)$
d) $x^2(x-5)(x-9)$
e) $2x(7x - 15)$
f) $x^2(x+4)(x-2)$
g) $x^3(x^3 - 18x + 72)$
h) $x^4(x - 91)$
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and science. For example, in physics, you might use factoring to solve equations that describe the motion of an object. In engineering, factoring can help simplify complex expressions when designing structures or circuits. In computer science, factoring is used in cryptography to break codes and secure data.
