1. **Problem Statement:** Rod is thinking of a polynomial in one variable with these characteristics: - Degree 4 - Three terms - One term of degree 1 - One term with coefficient -5 - Y-intercept at 10 2. **Understanding the problem:** - A polynomial of degree 4 means the highest power of the variable is 4. - Three terms means the polynomial looks like $ax^4 + bx + c$ or similar, but must include a term of degree 1. - One term has coefficient -5, so one of $a$, $b$, or $c$ is -5. - Y-intercept at 10 means when $x=0$, $y=10$, so the constant term $c=10$. 3. **Form of the polynomial:** Let the polynomial be: $$f(x) = ax^4 + bx + c$$ where $a$, $b$, and $c$ are constants. 4. **Apply conditions:** - Degree 4 means $a \neq 0$. - Three terms: $ax^4$, $bx$, and $c$. - One term has degree 1: $bx$ term. - One coefficient is -5: either $a = -5$, or $b = -5$, or $c = -5$. - Y-intercept at 10 means $f(0) = c = 10$. Since $c=10$, coefficient -5 cannot be $c$. 5. **Possible cases for coefficient -5:** - Case 1: $a = -5$ - Case 2: $b = -5$ 6. **Construct polynomials:** - Case 1: $f(x) = -5x^4 + bx + 10$, with $b$ any real number except 0 (to keep degree 1 term). - Case 2: $f(x) = ax^4 - 5x + 10$, with $a \neq 0$. 7. **Examples:** - Example 1: $f(x) = -5x^4 + 3x + 10$ - Example 2: $f(x) = 2x^4 - 5x + 10$ 8. **Are these the only possible answers?** No, because $a$ and $b$ can be any real numbers satisfying the conditions above, so infinitely many polynomials fit the description. **Final answer:** Two example polynomials are: $$f(x) = -5x^4 + 3x + 10$$ $$g(x) = 2x^4 - 5x + 10$$ These are not the only possible polynomials; any polynomial of the form $ax^4 + bx + 10$ with $a \neq 0$, $b \neq 0$, and either $a = -5$ or $b = -5$ satisfies the conditions.