We appreciate your visit to 10 Let s factorize the following expressions a tex x 2 15x 2 56x tex b tex x 3 17x 2 42x tex c tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
- Combine like terms where possible.
- Factor out the greatest common factor (GCF).
- Factor quadratic expressions into two binomials.
- 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. Understanding the Problem
We are asked to factorise the given polynomial expressions. This involves finding common factors and expressing each polynomial as a product of simpler expressions.
2. Factoring a)
a) $x^2+15 x^2+56 x$
First, combine like terms: $x^2 + 15x^2 = 16x^2$. So the expression becomes $16x^2 + 56x$.
Now, factor out the common factor $x$: $16x^2 + 56x = x(16x + 56)$.
Next, factor out the greatest common divisor of 16 and 56, which is 8: $x(16x + 56) = 8x(2x + 7)$.
3. Factoring b)
b) $x^3+17 x^2+42 x$
Factor out the common factor $x$: $x^3 + 17x^2 + 42x = x(x^2 + 17x + 42)$.
Now, factor the quadratic expression $x^2 + 17x + 42$. We look for two numbers that multiply to 42 and add to 17. These numbers are 3 and 14. So, $x^2 + 17x + 42 = (x+3)(x+14)$.
Therefore, $x^3 + 17x^2 + 42x = x(x+3)(x+14)$.
4. Factoring c)
c) $x^4-4 x^4-21 x^2$
First, combine like terms: $x^4 - 4x^4 = -3x^4$. So the expression becomes $-3x^4 - 21x^2$.
Now, factor out the common factor $-3x^2$: $-3x^4 - 21x^2 = -3x^2(x^2 + 7)$.
5. Factoring d)
d) $x^4-14 x^3+45 x^2$
Factor out the common factor $x^2$: $x^4 - 14x^3 + 45x^2 = x^2(x^2 - 14x + 45)$.
Now, factor the quadratic expression $x^2 - 14x + 45$. We look for two numbers that multiply to 45 and add to -14. These numbers are -5 and -9. So, $x^2 - 14x + 45 = (x-5)(x-9)$.
Therefore, $x^4 - 14x^3 + 45x^2 = x^2(x-5)(x-9)$.
6. Factoring e)
e) $x^2+13 x^2-30 x$
First, combine like terms: $x^2 + 13x^2 = 14x^2$. So the expression becomes $14x^2 - 30x$.
Now, factor out the common factor $x$: $14x^2 - 30x = x(14x - 30)$.
Next, factor out the greatest common divisor of 14 and 30, which is 2: $x(14x - 30) = 2x(7x - 15)$.
7. Factoring f)
f) $x^4+2 x^3-8 x^2$
Factor out the common factor $x^2$: $x^4 + 2x^3 - 8x^2 = x^2(x^2 + 2x - 8)$.
Now, factor the quadratic expression $x^2 + 2x - 8$. We look for two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. So, $x^2 + 2x - 8 = (x+4)(x-2)$.
Therefore, $x^4 + 2x^3 - 8x^2 = x^2(x+4)(x-2)$.
8. Factoring g)
g) $x^6-18 x^4+72 x^3$
Factor out the common factor $x^3$: $x^6 - 18x^4 + 72x^3 = x^3(x^3 - 18x + 72)$.
9. Factoring h)
h) $x^5+11 x^4-102 x^4$
First, combine like terms: $11x^4 - 102x^4 = -91x^4$. So the expression becomes $x^5 - 91x^4$.
Now, 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 extensively in various fields. For example, in engineering, factoring can help simplify complex equations that model physical systems, making them easier to analyze and solve. In computer graphics, factoring can be used to optimize calculations for rendering images, improving performance and efficiency. Moreover, in economics, factoring can be applied to analyze cost functions and optimize production processes, leading to better resource allocation and increased profitability.
- Factor out the greatest common factor (GCF).
- Factor quadratic expressions into two binomials.
- 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. Understanding the Problem
We are asked to factorise the given polynomial expressions. This involves finding common factors and expressing each polynomial as a product of simpler expressions.
2. Factoring a)
a) $x^2+15 x^2+56 x$
First, combine like terms: $x^2 + 15x^2 = 16x^2$. So the expression becomes $16x^2 + 56x$.
Now, factor out the common factor $x$: $16x^2 + 56x = x(16x + 56)$.
Next, factor out the greatest common divisor of 16 and 56, which is 8: $x(16x + 56) = 8x(2x + 7)$.
3. Factoring b)
b) $x^3+17 x^2+42 x$
Factor out the common factor $x$: $x^3 + 17x^2 + 42x = x(x^2 + 17x + 42)$.
Now, factor the quadratic expression $x^2 + 17x + 42$. We look for two numbers that multiply to 42 and add to 17. These numbers are 3 and 14. So, $x^2 + 17x + 42 = (x+3)(x+14)$.
Therefore, $x^3 + 17x^2 + 42x = x(x+3)(x+14)$.
4. Factoring c)
c) $x^4-4 x^4-21 x^2$
First, combine like terms: $x^4 - 4x^4 = -3x^4$. So the expression becomes $-3x^4 - 21x^2$.
Now, factor out the common factor $-3x^2$: $-3x^4 - 21x^2 = -3x^2(x^2 + 7)$.
5. Factoring d)
d) $x^4-14 x^3+45 x^2$
Factor out the common factor $x^2$: $x^4 - 14x^3 + 45x^2 = x^2(x^2 - 14x + 45)$.
Now, factor the quadratic expression $x^2 - 14x + 45$. We look for two numbers that multiply to 45 and add to -14. These numbers are -5 and -9. So, $x^2 - 14x + 45 = (x-5)(x-9)$.
Therefore, $x^4 - 14x^3 + 45x^2 = x^2(x-5)(x-9)$.
6. Factoring e)
e) $x^2+13 x^2-30 x$
First, combine like terms: $x^2 + 13x^2 = 14x^2$. So the expression becomes $14x^2 - 30x$.
Now, factor out the common factor $x$: $14x^2 - 30x = x(14x - 30)$.
Next, factor out the greatest common divisor of 14 and 30, which is 2: $x(14x - 30) = 2x(7x - 15)$.
7. Factoring f)
f) $x^4+2 x^3-8 x^2$
Factor out the common factor $x^2$: $x^4 + 2x^3 - 8x^2 = x^2(x^2 + 2x - 8)$.
Now, factor the quadratic expression $x^2 + 2x - 8$. We look for two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. So, $x^2 + 2x - 8 = (x+4)(x-2)$.
Therefore, $x^4 + 2x^3 - 8x^2 = x^2(x+4)(x-2)$.
8. Factoring g)
g) $x^6-18 x^4+72 x^3$
Factor out the common factor $x^3$: $x^6 - 18x^4 + 72x^3 = x^3(x^3 - 18x + 72)$.
9. Factoring h)
h) $x^5+11 x^4-102 x^4$
First, combine like terms: $11x^4 - 102x^4 = -91x^4$. So the expression becomes $x^5 - 91x^4$.
Now, 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 extensively in various fields. For example, in engineering, factoring can help simplify complex equations that model physical systems, making them easier to analyze and solve. In computer graphics, factoring can be used to optimize calculations for rendering images, improving performance and efficiency. Moreover, in economics, factoring can be applied to analyze cost functions and optimize production processes, leading to better resource allocation and increased profitability.
Thanks for taking the time to read 10 Let s factorize the following expressions a tex x 2 15x 2 56x tex b tex x 3 17x 2 42x tex c tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
