A polynomial $f ( x )$ with real coefficients is said to be a sum of squares if we can write $f ( x ) = p _ { 1 } ( x ) ^ { 2 } + \cdots + p _ { k } ( x ) ^ { 2 }$, where $p _ { 1 } ( x ) , \ldots , p _ { k } ( x )$ are polynomials with real coefficients. For each statement below, write whether it is TRUE or FALSE. a) If a polynomial $f ( x )$ is a sum of squares, then the coefficient of every odd power of $x$ in $f ( x )$ must be 0. Answer: $\_\_\_\_$ b) If $f ( x ) = x ^ { 2 } + p x + q$ has a non-real root, then $f ( x )$ is a sum of squares. Answer: $\_\_\_\_$ c) If $f ( x ) = x ^ { 3 } + p x ^ { 2 } + q x + r$ has a non-real root, then $f ( x )$ is a sum of squares. Answer: $\_\_\_\_$ d) If a polynomial $f ( x ) > 0$ for all real values of $x$, then $f ( x )$ is a sum of squares. Answer: $\_\_\_\_$
A polynomial $f ( x )$ with real coefficients is said to be a sum of squares if we can write $f ( x ) = p _ { 1 } ( x ) ^ { 2 } + \cdots + p _ { k } ( x ) ^ { 2 }$, where $p _ { 1 } ( x ) , \ldots , p _ { k } ( x )$ are polynomials with real coefficients. For each statement below, write whether it is TRUE or FALSE.\\
a) If a polynomial $f ( x )$ is a sum of squares, then the coefficient of every odd power of $x$ in $f ( x )$ must be 0.
Answer: $\_\_\_\_$\\
b) If $f ( x ) = x ^ { 2 } + p x + q$ has a non-real root, then $f ( x )$ is a sum of squares.
Answer: $\_\_\_\_$\\
c) If $f ( x ) = x ^ { 3 } + p x ^ { 2 } + q x + r$ has a non-real root, then $f ( x )$ is a sum of squares.
Answer: $\_\_\_\_$\\
d) If a polynomial $f ( x ) > 0$ for all real values of $x$, then $f ( x )$ is a sum of squares.
Answer: $\_\_\_\_$