Question

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chapter review
what is the degree of the polynomials?
1 (28x^4 + 3x^2) 2 (a^7 + a^4b^6) 3 (x^2 - 3y^2x^3 + y^3)

Explanation:

Problem 1: \( 28x^4 + 3x^2 \)

Step 1: Recall the definition of the degree of a polynomial

The degree of a polynomial is the highest power (exponent) of the variable in the polynomial.

Step 2: Identify the exponents of each term

For the term \( 28x^4 \), the exponent of \( x \) is \( 4 \). For the term \( 3x^2 \), the exponent of \( x \) is \( 2 \).

Step 3: Determine the highest exponent

Comparing \( 4 \) and \( 2 \), the highest exponent is \( 4 \).

Step 1: Recall the definition for polynomials with multiple variables

For a term with multiple variables, the degree of the term is the sum of the exponents of the variables. The degree of the polynomial is the highest degree among its terms.

Step 2: Find the degree of each term

• For the term \( a^7 \), the degree is \( 7 \) (only one variable \( a \) with exponent \( 7 \)).
• For the term \( a^4b^6 \), the degree is \( 4 + 6 = 10 \) (sum of exponents of \( a \) and \( b \)).

Step 3: Determine the highest degree

Comparing \( 7 \) and \( 10 \), the highest degree is \( 10 \).

Step 1: Recall the definition for polynomials with multiple variables

As before, for a term with multiple variables, the degree is the sum of the exponents of the variables. The degree of the polynomial is the highest degree among its terms.

Step 2: Find the degree of each term

• For the term \( x^2 \), the degree is \( 2 \) (exponent of \( x \) is \( 2 \)).
• For the term \( -3y^2x^3 \), the degree is \( 2 + 3 = 5 \) (sum of exponents of \( y \) and \( x \)).
• For the term \( y^3 \), the degree is \( 3 \) (exponent of \( y \) is \( 3 \)).

Step 3: Determine the highest degree

Comparing \( 2 \), \( 5 \), and \( 3 \), the highest degree is \( 5 \).

Answer:

The degree of the polynomial \( 28x^4 + 3x^2 \) is \( 4 \).

Problem 2: \( a^7 + a^4b^6 \)