Answer :
Sure! Let's break down the solution for each part of the question, factoring the polynomials step by step:
### Part (a): Factor [tex]\( x^2 - 2x - 35 \)[/tex]
1. Identify a, b, c: The given quadratic is in the form [tex]\( ax^2 + bx + c \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -35 \)[/tex].
2. Factor by finding two numbers that multiply to -35 and add to -2: We need two numbers that multiply to [tex]\(-35\)[/tex] and add up to [tex]\(-2\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(5\)[/tex].
3. Factor the quadratic: This gives us:
[tex]\[
(x - 7)(x + 5)
\][/tex]
### Part (b): Factor [tex]\( 7x^2 + 13x - 24 \)[/tex]
1. Identify a, b, c: The quadratic is [tex]\( ax^2 + bx + c \)[/tex] with [tex]\( a = 7 \)[/tex], [tex]\( b = 13 \)[/tex], and [tex]\( c = -24 \)[/tex].
2. Use the factored form by finding numbers: We look for two numbers that multiply to [tex]\( a \times c = 7 \times -24 = -168 \)[/tex] and add to [tex]\( b = 13\)[/tex].
3. Find the numbers: The numbers that work are [tex]\( 21 \)[/tex] and [tex]\(-8\)[/tex] (because [tex]\( 21 \times -8 = -168\)[/tex] and [tex]\( 21 - 8 = 13\)[/tex]).
4. Rewrite and factor by grouping:
[tex]\[
7x^2 + 21x - 8x - 24
\][/tex]
Group and factor:
[tex]\[
(7x^2 + 21x) + (-8x - 24)
\][/tex]
Factor out the common factors:
[tex]\[
7x(x + 3) - 8(x + 3)
\][/tex]
5. Since [tex]\( (x + 3) \)[/tex] is common, factor it out:
[tex]\[
(x + 3)(7x - 8)
\][/tex]
### Part (c): Factor [tex]\( x^6 - 2x^5 - 15x^4 \)[/tex]
1. Factor out the greatest common factor (GCF): Notice that each term has at least [tex]\( x^4 \)[/tex].
[tex]\[
x^4(x^2 - 2x - 15)
\][/tex]
2. Factor the quadratic inside the parentheses: Now deal with [tex]\( x^2 - 2x - 15 \)[/tex].
- Find numbers that multiply to [tex]\(-15\)[/tex] and add to [tex]\(-2\)[/tex]. They are [tex]\(-5\)[/tex] and [tex]\(3\)[/tex].
- Factor the quadratic:
[tex]\[
(x - 5)(x + 3)
\][/tex]
3. Put it all together:
[tex]\[
x^4(x - 5)(x + 3)
\][/tex]
These steps detail the complete factorization of each polynomial. If you have any further questions or need additional clarification on any step, feel free to ask!
### Part (a): Factor [tex]\( x^2 - 2x - 35 \)[/tex]
1. Identify a, b, c: The given quadratic is in the form [tex]\( ax^2 + bx + c \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -35 \)[/tex].
2. Factor by finding two numbers that multiply to -35 and add to -2: We need two numbers that multiply to [tex]\(-35\)[/tex] and add up to [tex]\(-2\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(5\)[/tex].
3. Factor the quadratic: This gives us:
[tex]\[
(x - 7)(x + 5)
\][/tex]
### Part (b): Factor [tex]\( 7x^2 + 13x - 24 \)[/tex]
1. Identify a, b, c: The quadratic is [tex]\( ax^2 + bx + c \)[/tex] with [tex]\( a = 7 \)[/tex], [tex]\( b = 13 \)[/tex], and [tex]\( c = -24 \)[/tex].
2. Use the factored form by finding numbers: We look for two numbers that multiply to [tex]\( a \times c = 7 \times -24 = -168 \)[/tex] and add to [tex]\( b = 13\)[/tex].
3. Find the numbers: The numbers that work are [tex]\( 21 \)[/tex] and [tex]\(-8\)[/tex] (because [tex]\( 21 \times -8 = -168\)[/tex] and [tex]\( 21 - 8 = 13\)[/tex]).
4. Rewrite and factor by grouping:
[tex]\[
7x^2 + 21x - 8x - 24
\][/tex]
Group and factor:
[tex]\[
(7x^2 + 21x) + (-8x - 24)
\][/tex]
Factor out the common factors:
[tex]\[
7x(x + 3) - 8(x + 3)
\][/tex]
5. Since [tex]\( (x + 3) \)[/tex] is common, factor it out:
[tex]\[
(x + 3)(7x - 8)
\][/tex]
### Part (c): Factor [tex]\( x^6 - 2x^5 - 15x^4 \)[/tex]
1. Factor out the greatest common factor (GCF): Notice that each term has at least [tex]\( x^4 \)[/tex].
[tex]\[
x^4(x^2 - 2x - 15)
\][/tex]
2. Factor the quadratic inside the parentheses: Now deal with [tex]\( x^2 - 2x - 15 \)[/tex].
- Find numbers that multiply to [tex]\(-15\)[/tex] and add to [tex]\(-2\)[/tex]. They are [tex]\(-5\)[/tex] and [tex]\(3\)[/tex].
- Factor the quadratic:
[tex]\[
(x - 5)(x + 3)
\][/tex]
3. Put it all together:
[tex]\[
x^4(x - 5)(x + 3)
\][/tex]
These steps detail the complete factorization of each polynomial. If you have any further questions or need additional clarification on any step, feel free to ask!
