Question

1. classify the following polynomials by degree and number of terms.

a) 3x + 1
b) 5x²
c) x⁴ - 2x + 8
d) 10
e) x³ - 5x² + 7x - 3
a.)
b.)
c.)
d.)
e.)

Explanation:

Step1: Recall Definitions

• Degree of a polynomial: The highest power of the variable.
• Number of terms: Count of monomials (terms) separated by + or - signs.
• Classification by degree: Constant (degree 0), Linear (degree 1), Quadratic (degree 2), Cubic (degree 3), Quartic (degree 4), etc.
• Classification by terms: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms), Polynomial (≥4 terms).

Step2: Analyze Part (a) - \(3x + 1\)

• Degree: Highest power of \(x\) is \(1\) (linear).
• Number of terms: Two terms (\(3x\) and \(1\)) → binomial.
• Classification: Linear binomial.

Step3: Analyze Part (b) - \(5x^2\)

• Degree: Highest power of \(x\) is \(2\) (quadratic).
• Number of terms: One term → monomial.
• Classification: Quadratic monomial.

Step4: Analyze Part (c) - \(x^4 - 2x + 8\)

• Degree: Highest power of \(x\) is \(4\) (quartic).
• Number of terms: Three terms (\(x^4\), \(-2x\), \(8\)) → trinomial.
• Classification: Quartic trinomial.

Step5: Analyze Part (d) - \(10\)

• Degree: Highest power of \(x\) (implied \(x^0\)) is \(0\) (constant).
• Number of terms: One term → monomial.
• Classification: Constant monomial.

Step6: Analyze Part (e) - \(x^3 - 5x^2 + 7x - 3\)

• Degree: Highest power of \(x\) is \(3\) (cubic).
• Number of terms: Four terms (\(x^3\), \(-5x^2\), \(7x\), \(-3\)) → polynomial (or cubic polynomial with 4 terms).
• Classification: Cubic polynomial (or cubic quadrinomial, but "cubic polynomial" is common for ≥4 terms).

Answer:

a) Linear binomial
b) Quadratic monomial
c) Quartic trinomial
d) Constant monomial
e) Cubic polynomial (or Cubic quadrinomial)