Question

Consider the following polynomial. - 2x * y ^ 5 + 3x ^ 7 - 10x ^ 3 * y ^ 6 + 8x ^ 6 + 4 Which of the following statements are true? Select all that apply. polynomial has 5 terms. The leading coefficient is -2 degree of the polynomial is 7. The constant term is 4. polynomial is in decreasing degree order

Answer 1

Option a: polynomial has 5 terms.

Option c: The constant term is 4.

Explanation:

The polynomial is $-2xy^{5} +3x^{7} -10x^{3} y^{6} +8x^{6} +4$

Option a: Polynomial has 5 terms

From the polynomial, we have,

1st term = $-2 x y^{5}$

2nd term = $3 x^{7}$

3rd term = $-10 x^{3} y^{6}$

4th term = $8 x^{6}$

5th term = $4$

Hence, the polynomial has 5 terms.

Thus, Option a is the correct answer.

Option b: The leading coefficient is -2 and the degree of the polynomial is 7.

The leading coefficient is the coefficient of the variable with the largest degree.

The degree of the polynomial can be determine by the adding the powers of the variable.

degree of 1st term = $-2 x y^{5}$ = $1+5=6$

degree of 2nd term = $3 x^{7}$ $=7$

degree of 3rd term = $-10 x^{3} y^{6}$ $=3+6=9$

degree of 4th term = $8 x^{6}$ $=6$

degree of 5th term = $4$ $=0$

Thus, from these 5 terms the highest value is 9 which is the degree of the polynomial.

The polynomial $-2xy^{5} +3x^{7} -10x^{3} y^{6} +8x^{6} +4$ has a largest degree of 9 and the leading coefficient is -10.

Hence, Option b is not the correct answer.

Option c: The constant term is 4.

The constant term is a term in which variable will not occur.

From the polynomial $-2xy^{5} +3x^{7} -10x^{3} y^{6} +8x^{6} +4$, we can see that the constant term is 4.

Hence, Option c is the correct answer.

Option d: Polynomial is in decreasing degree order

The decreasing degree order is the degree in which each term is no larger than the degree of the preceding term.

From the polynomial, we have,

degree of 1st term = $-2 x y^{5}$ = $1+5=6$

degree of 2nd term = $3 x^{7}$ $=7$

degree of 3rd term = $-10 x^{3} y^{6}$ $=3+6=9$

degree of 4th term = $8 x^{6}$ $=6$

degree of 5th term = $4$ $=0$

where the degree of the 3rd term is not less than the 4th term.

Thus, the polynomial is not in decreasing degree order.

Hence, Option d is not the correct answer.