The product of two binomials, u (x) and v (x), is a binomial of degree 8 with a positive leading coefficient. What must be true about the sum of u (x) and v (x) ?

From the above conclusion, it can be inferred that sum of u (x) and v (x) is a binomial of degree 4 with a positive or negative leading coefficient depending on the value of the constant 'a'

Step-by-step explanation:

If the product of two binomials, u (x) and v (x) has a binomial of degree 8, then the individual binomial will have a binomial of degree 4. For example, lets assume u (x) = ax⁴ + bx and v (x) ax⁴+cx. Note that they are both functions of x and contain only two terms since they are binomial.

Taking their product we have;

u (x) * v (x) = (ax⁴ + bx) * (ax⁴+cx)

u (x) * v (x) = a²x⁸+acx⁵+abx⁵+bcx²

it can be seen that the higest power of x is 8 which gives the degree of the product.

Taking their sum;

u (x) + v (x) = (ax⁴ + bx) + (ax⁴+cx)

u (x) + v (x) = ax⁴ + ax⁴ + bx + cx

u (x) + v (x) = 2ax⁴ + bx + cx

It can be seen that the digest power of x is 4 which gives the degree of the sum of the binomial.

From the above conclusion, it can be inferred that sum of u (x) and v (x) is a binomial of degree 4 with a positive or negative leading coefficient depending on the value of the constant 'a'