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Sagot :
Sure! Let's fill in the table by identifying the Greatest Common Factor (GCF) and then writing the polynomial in its factored form.
[tex]\[ \begin{array}{|l|c|c|c|} \hline \text{Polynomial} & \text{GCF} & \text{Quotient of Polynomial} & \text{Factored Form} \\ \hline x^2 + 2x & x & x + 2 & x(x + 2) \\ \hline 14t^2 + 35tx & 7t & 2t + 5x & 7t(2t + 5x) \\ \hline 30h^2 - 6hy & 6h & 5h - y & 6h(5h - y) \\ \hline 5x^4 - 50x^3 & 5x^3 & x - 10 & 5x^3(x - 10) \\ \hline 6x^4 - 12x^2 - 9x & 3x & 2x^3 - 4x - 3 & 3x(2x^3 - 4x - 3) \\ \hline \end{array} \][/tex]
Here is the detailed step-by-step solution for each polynomial:
1. Polynomial: [tex]\(x^2 + 2x\)[/tex]
- GCF: [tex]\(x\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{x^2}{x} = x, \quad \frac{2x}{x} = 2 \][/tex]
So, the quotient is [tex]\(x + 2\)[/tex]
- Factored Form: [tex]\(x(x + 2)\)[/tex]
2. Polynomial: [tex]\(14t^2 + 35tx\)[/tex]
- GCF: [tex]\(7t\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{14t^2}{7t} = 2t, \quad \frac{35tx}{7t} = 5x \][/tex]
So, the quotient is [tex]\(2t + 5x\)[/tex]
- Factored Form: [tex]\(7t(2t + 5x)\)[/tex]
3. Polynomial: [tex]\(30h^2 - 6hy\)[/tex]
- GCF: [tex]\(6h\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{30h^2}{6h} = 5h, \quad \frac{6hy}{6h} = y \][/tex]
So, the quotient is [tex]\(5h - y\)[/tex]
- Factored Form: [tex]\(6h(5h - y)\)[/tex]
4. Polynomial: [tex]\(5x^4 - 50x^3\)[/tex]
- GCF: [tex]\(5x^3\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{5x^4}{5x^3} = x, \quad \frac{50x^3}{5x^3} = 10 \][/tex]
So, the quotient is [tex]\(x - 10\)[/tex]
- Factored Form: [tex]\(5x^3(x - 10)\)[/tex]
5. Polynomial: [tex]\(6x^4 - 12x^2 - 9x\)[/tex]
- GCF: [tex]\(3x\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{6x^4}{3x} = 2x^3, \quad \frac{12x^2}{3x} = 4x, \quad \frac{9x}{3x} = 3 \][/tex]
So, the quotient is [tex]\(2x^3 - 4x - 3\)[/tex]
- Factored Form: [tex]\(3x(2x^3 - 4x - 3)\)[/tex]
Now, the table is correctly filled in.
[tex]\[ \begin{array}{|l|c|c|c|} \hline \text{Polynomial} & \text{GCF} & \text{Quotient of Polynomial} & \text{Factored Form} \\ \hline x^2 + 2x & x & x + 2 & x(x + 2) \\ \hline 14t^2 + 35tx & 7t & 2t + 5x & 7t(2t + 5x) \\ \hline 30h^2 - 6hy & 6h & 5h - y & 6h(5h - y) \\ \hline 5x^4 - 50x^3 & 5x^3 & x - 10 & 5x^3(x - 10) \\ \hline 6x^4 - 12x^2 - 9x & 3x & 2x^3 - 4x - 3 & 3x(2x^3 - 4x - 3) \\ \hline \end{array} \][/tex]
Here is the detailed step-by-step solution for each polynomial:
1. Polynomial: [tex]\(x^2 + 2x\)[/tex]
- GCF: [tex]\(x\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{x^2}{x} = x, \quad \frac{2x}{x} = 2 \][/tex]
So, the quotient is [tex]\(x + 2\)[/tex]
- Factored Form: [tex]\(x(x + 2)\)[/tex]
2. Polynomial: [tex]\(14t^2 + 35tx\)[/tex]
- GCF: [tex]\(7t\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{14t^2}{7t} = 2t, \quad \frac{35tx}{7t} = 5x \][/tex]
So, the quotient is [tex]\(2t + 5x\)[/tex]
- Factored Form: [tex]\(7t(2t + 5x)\)[/tex]
3. Polynomial: [tex]\(30h^2 - 6hy\)[/tex]
- GCF: [tex]\(6h\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{30h^2}{6h} = 5h, \quad \frac{6hy}{6h} = y \][/tex]
So, the quotient is [tex]\(5h - y\)[/tex]
- Factored Form: [tex]\(6h(5h - y)\)[/tex]
4. Polynomial: [tex]\(5x^4 - 50x^3\)[/tex]
- GCF: [tex]\(5x^3\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{5x^4}{5x^3} = x, \quad \frac{50x^3}{5x^3} = 10 \][/tex]
So, the quotient is [tex]\(x - 10\)[/tex]
- Factored Form: [tex]\(5x^3(x - 10)\)[/tex]
5. Polynomial: [tex]\(6x^4 - 12x^2 - 9x\)[/tex]
- GCF: [tex]\(3x\)[/tex]
- Quotient of Polynomial: To find the quotient, divide each term by the GCF:
[tex]\[ \frac{6x^4}{3x} = 2x^3, \quad \frac{12x^2}{3x} = 4x, \quad \frac{9x}{3x} = 3 \][/tex]
So, the quotient is [tex]\(2x^3 - 4x - 3\)[/tex]
- Factored Form: [tex]\(3x(2x^3 - 4x - 3)\)[/tex]
Now, the table is correctly filled in.
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