Answer :
Let's factor each given expression step by step.
1. Factor the expression [tex]\( 16x^2 - 8x + 1 \)[/tex]:
To factor this quadratic expression, we first look for two numbers that multiply to [tex]\(16 \cdot 1 = 16\)[/tex] and add to [tex]\(-8\)[/tex]. Since this expression is a perfect square trinomial, it can be factored as:
[tex]\[ (4x - 1)^2 \][/tex]
Therefore,
[tex]\[ 16x^2 - 8x + 1 = (4x - 1)^2 \][/tex]
2. Factor the expression [tex]\( x^2 - x - 30 \)[/tex]:
To factor this quadratic expression, we need to find two numbers that multiply to [tex]\(-30\)[/tex] and add to [tex]\(-1\)[/tex]. The numbers [tex]\(-6\)[/tex] and [tex]\(5\)[/tex] fit these criteria. Hence, we can factor as:
[tex]\[ (x - 6)(x + 5) \][/tex]
Therefore,
[tex]\[ x^2 - x - 30 = (x - 6)(x + 5) \][/tex]
3. Factor the expression [tex]\( 9x^2 - 49 \)[/tex]:
This is a difference of squares. We can write:
[tex]\[ 9x^2 - 49 = (3x)^2 - 7^2 \][/tex]
Therefore, using the difference of squares formula, we have:
[tex]\[ 9x^2 - 49 = (3x - 7)(3x + 7) \][/tex]
4. Factor the expression [tex]\( 3x^2 + 17x - 6 \)[/tex]:
We need to find two numbers that multiply to [tex]\(3 \cdot -6 = -18\)[/tex] and add to [tex]\(17\)[/tex]. The pair [tex]\(18\)[/tex] and [tex]\(-1\)[/tex] fit this condition. We can rewrite the middle term using these numbers:
[tex]\[ 3x^2 + 18x - x - 6 \][/tex]
Now, factor by grouping:
[tex]\[ 3x(x + 6) - 1(x + 6) \][/tex]
Factor out the common factor [tex]\((x + 6)\)[/tex]:
[tex]\[ (x + 6)(3x - 1) \][/tex]
Therefore,
[tex]\[ 3x^2 + 17x - 6 = (x + 6)(3x - 1) \][/tex]
Putting it all together, we have the factored forms of each expression. Thus:
[tex]\[ \begin{array}{ll} 16 x^2-8 x+1 & = (4x - 1)^2 \\ x^2-x-30 & = (x - 6)(x + 5) \\ 9 x^2-49 & = (3x - 7)(3x + 7) \\ 3 x^2+17 x-6 & = (x + 6)(3x - 1) \\ \end{array} \][/tex]
1. Factor the expression [tex]\( 16x^2 - 8x + 1 \)[/tex]:
To factor this quadratic expression, we first look for two numbers that multiply to [tex]\(16 \cdot 1 = 16\)[/tex] and add to [tex]\(-8\)[/tex]. Since this expression is a perfect square trinomial, it can be factored as:
[tex]\[ (4x - 1)^2 \][/tex]
Therefore,
[tex]\[ 16x^2 - 8x + 1 = (4x - 1)^2 \][/tex]
2. Factor the expression [tex]\( x^2 - x - 30 \)[/tex]:
To factor this quadratic expression, we need to find two numbers that multiply to [tex]\(-30\)[/tex] and add to [tex]\(-1\)[/tex]. The numbers [tex]\(-6\)[/tex] and [tex]\(5\)[/tex] fit these criteria. Hence, we can factor as:
[tex]\[ (x - 6)(x + 5) \][/tex]
Therefore,
[tex]\[ x^2 - x - 30 = (x - 6)(x + 5) \][/tex]
3. Factor the expression [tex]\( 9x^2 - 49 \)[/tex]:
This is a difference of squares. We can write:
[tex]\[ 9x^2 - 49 = (3x)^2 - 7^2 \][/tex]
Therefore, using the difference of squares formula, we have:
[tex]\[ 9x^2 - 49 = (3x - 7)(3x + 7) \][/tex]
4. Factor the expression [tex]\( 3x^2 + 17x - 6 \)[/tex]:
We need to find two numbers that multiply to [tex]\(3 \cdot -6 = -18\)[/tex] and add to [tex]\(17\)[/tex]. The pair [tex]\(18\)[/tex] and [tex]\(-1\)[/tex] fit this condition. We can rewrite the middle term using these numbers:
[tex]\[ 3x^2 + 18x - x - 6 \][/tex]
Now, factor by grouping:
[tex]\[ 3x(x + 6) - 1(x + 6) \][/tex]
Factor out the common factor [tex]\((x + 6)\)[/tex]:
[tex]\[ (x + 6)(3x - 1) \][/tex]
Therefore,
[tex]\[ 3x^2 + 17x - 6 = (x + 6)(3x - 1) \][/tex]
Putting it all together, we have the factored forms of each expression. Thus:
[tex]\[ \begin{array}{ll} 16 x^2-8 x+1 & = (4x - 1)^2 \\ x^2-x-30 & = (x - 6)(x + 5) \\ 9 x^2-49 & = (3x - 7)(3x + 7) \\ 3 x^2+17 x-6 & = (x + 6)(3x - 1) \\ \end{array} \][/tex]
