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 (say $x$) is 4. - Three terms means the polynomial looks like $ax^4 + bx + c$ or similar, but must have exactly three terms. - One term has degree 1 means there is a term with $x^1$. - One term has coefficient -5 means one of the coefficients is -5. - y-intercept at 10 means when $x=0$, the polynomial equals 10, so the constant term is 10. 3. **Form of the polynomial:** Let the polynomial be: $$p(x) = ax^4 + bx + c$$ where $a$, $b$, and $c$ are constants. 4. **Apply conditions:** - Degree 4 means $a \neq 0$. - Three terms: $ax^4$, $bx$, and $c$ (constant term). - 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 $p(0) = c = 10$. Since $c=10$, the coefficient -5 must be either $a$ or $b$. 5. **Two possible polynomials:** - Case 1: $a = -5$, $b$ any real number except 0 (to keep three terms), $c=10$: $$p_1(x) = -5x^4 + bx + 10$$ - Case 2: $b = -5$, $a \neq 0$, $c=10$: $$p_2(x) = ax^4 - 5x + 10$$ 6. **Are these the only possible answers?** No, because $a$ and $b$ can be any real numbers satisfying the above conditions, so there are infinitely many polynomials fitting the criteria. **Final answer:** Two example polynomials are: $$p_1(x) = -5x^4 + 3x + 10$$ $$p_2(x) = 2x^4 - 5x + 10$$ These satisfy all the conditions. There are infinitely many such polynomials because $a$ and $b$ can vary as long as the conditions hold.