Question Find a 3-term polynomial with $x^{2}$ as the greatest common factor. Answers may vary.

Studdy Solution

STEP 1

1. A polynomial with three terms is of the form $ax^n + bx^m + cx^k$ .
2. The greatest common factor (GCF) of the polynomial terms must be $x^2$ .
3. The exponents $n$ , $m$ , and $k$ must be integers greater than or equal to 2 since $x^2$ is the GCF.
4. The coefficients $a$ , $b$ , and $c$ can be any real numbers.

STEP 2

1. Choose exponents for $x$ that are greater than or equal to 2.
2. Choose coefficients for each term.
3. Write the polynomial.

STEP 3

Choose exponents for $x$ in each term that are greater than or equal to 2. For simplicity, let's choose $n=2$ , $m=3$ , and $k=4$ .

STEP 4

Choose coefficients for each term. Let's choose $a=3$ , $b=5$ , and $c=7$ .

STEP 5

Write the polynomial using the chosen exponents and coefficients. The polynomial is:
$3x^2 + 5x^3 + 7x^4$
This polynomial has a GCF of $x^2$ because each term is divisible by $x^2$ .
The solution is the polynomial $3x^2 + 5x^3 + 7x^4$ .

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