Question

math 2 chapter 2.4 factoring polynomials
factor the gcf out of each polynomial.

1. \\( 10x^3 + 4x^2 + 8x \\) 2) \\( 3x^5y + 27x^3y^2 \\) 3) \\( 7x^7 + 14x^4 + 28x \\)

factor each polynomial completely.

1. \\( x^2 + 8x + 7 \\) 5) \\( x^2 + 10x + 21 \\) 6) \\( x^2 + 9x + 20 \\)
2. \\( x^2 + 11x + 30 \\) 8) \\( x^2 - 10x + 24 \\) 9) \\( x^2 + 2x - 48 \\)
3. \\( 6x^2 - 5x + 1 \\) 11) \\( 2x^2 + 5x + 2 \\) 12) \\( 4x^2 + 4x - 3 \\)
4. \\( 3x^2 + 3x - 6 \\) 14) \\( 8x^2 + 8x - 48 \\) 15) \\( 4x^2 - 19x - 5 \\)

Explanation:

Let's solve problem 1: \(10x^{3}+4x^{2}+8x\)

Step 1: Find the GCF of coefficients and variables

• For coefficients \(10\), \(4\), \(8\): The GCF of \(10\), \(4\), \(8\) is \(2\) (since \(10 = 2\times5\), \(4=2\times2\), \(8 = 2\times4\)).
• For variables \(x^{3}\), \(x^{2}\), \(x\): The GCF is \(x\) (using the lowest power of \(x\) which is \(x^{1}=x\)).
• So the GCF of the polynomial \(10x^{3}+4x^{2}+8x\) is \(2x\).

Step 2: Divide each term by the GCF

• Divide \(10x^{3}\) by \(2x\): \(\frac{10x^{3}}{2x}=5x^{2}\)
• Divide \(4x^{2}\) by \(2x\): \(\frac{4x^{2}}{2x} = 2x\)
• Divide \(8x\) by \(2x\): \(\frac{8x}{2x}=4\)

Step 3: Write the factored form

• Using the distributive property \(ab+ac+ad=a(b + c + d)\), where \(a = 2x\), \(b=5x^{2}\), \(c = 2x\), \(d = 4\), we get \(2x(5x^{2}+2x + 4)\)

Let's solve problem 4: \(x^{2}+8x + 7\)

Step 1: Find two numbers

We need two numbers that multiply to \(7\) (the constant term) and add up to \(8\) (the coefficient of \(x\)). The numbers \(1\) and \(7\) satisfy this because \(1\times7 = 7\) and \(1 + 7=8\).

Step 2: Factor the polynomial

Using the formula \(x^{2}+(a + b)x+ab=(x + a)(x + b)\), where \(a = 1\) and \(b = 7\), we get \((x + 1)(x + 7)\)

Let's solve problem 10: \(6x^{2}-5x + 1\)

Step 1: Multiply \(a\) and \(c\)

For the quadratic \(ax^{2}+bx + c\) (here \(a = 6\), \(b=- 5\), \(c = 1\)), we calculate \(a\times c=6\times1 = 6\).

Step 2: Find two numbers

We need two numbers that multiply to \(6\) and add up to \(b=-5\). The numbers \(-2\) and \(-3\) work because \((-2)\times(-3)=6\) and \(-2+(-3)=-5\).

Step 3: Rewrite the middle term

Rewrite \(-5x\) as \(-2x-3x\), so the polynomial becomes \(6x^{2}-2x-3x + 1\).

Step 4: Group and factor

• Group the first two terms and the last two terms: \((6x^{2}-2x)+(-3x + 1)\)
• Factor out GCF from each group: \(2x(3x - 1)-1(3x - 1)\)
• Now factor out \((3x - 1)\): \((2x - 1)(3x - 1)\)

Answer:

\(2x(5x^{2} + 2x + 4)\)