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Simplify Rational Function and Classify Polynomial Function: $F(x)=\\frac{31x^2 + 7x + 4}{19x^3 + 2x + 3x^2}$ and $g(x)=21x^4 + 5x + 2$
Mathematics
Grade 10 of Junior High School
Question Content
Simplify the rational function $F(x)=\\frac{31x^2 + 7x + 4}{19x^3 + 2x + 3x^2}$ and identify the form of polynomial function $g(x)=21x^4 + 5x + 2$
Correct Answer
1. $F(x)=\\frac{31x^2 + 7x + 4}{x(19x^2+3x+2)}$ (cannot be further simplified); 2. $g(x)$ is a quartic (4th-degree) trinomial polynomial function
Detailed Solution Steps
1
Step 1: Simplify $F(x)$: First, rearrange and factor the denominator of $F(x)$. Rearrange the denominator terms by degree: $19x^3 + 3x^2 + 2x$. Factor out the greatest common factor (GCF) $x$ from the denominator: $x(19x^2 + 3x + 2)$
2
Step 2: Check if the numerator $31x^2 + 7x + 4$ can be factored or shares common factors with the denominator. Use the discriminant of the quadratic numerator: $\\Delta = 7^2 - 4*31*4 = 49 - 496 = -447 < 0$, so the numerator has no real roots and cannot be factored over real numbers. There are no common factors between the numerator and denominator, so $F(x)$ is fully simplified as $\\frac{31x^2 + 7x + 4}{x(19x^2+3x+2)}$
3
Step 3: Analyze $g(x)$: Identify the highest power of $x$ in $g(x)=21x^4 + 5x + 2$, which is 4, so it is a 4th-degree (quartic) polynomial. Count the number of terms: there are 3 distinct terms, so it is a trinomial
Knowledge Points Involved
1
Simplification of Rational Functions
A rational function is a ratio of two polynomials. To simplify it, factor both the numerator and denominator completely, then cancel out any common non-zero factors. If the numerator or denominator is a quadratic with a negative discriminant, it cannot be factored over real numbers, so the function is in simplest form if no common factors exist.
2
Factoring Polynomials
Factoring involves rewriting a polynomial as a product of simpler polynomials. For polynomials with a common monomial factor, factor out the greatest common factor (GCF) first. For quadratics $ax^2+bx+c$, use the discriminant $\\Delta=b^2-4ac$ to determine if it can be factored over real numbers: if $\\Delta \\geq 0$, it has real roots and can be factored; if $\\Delta < 0$, it cannot be factored over real numbers.
3
Degree and Classification of Polynomials
The degree of a polynomial is the highest exponent of the variable in the polynomial. A polynomial is classified by its degree: 1st-degree (linear), 2nd-degree (quadratic), 3rd-degree (cubic), 4th-degree (quartic), etc. It is also classified by the number of terms: monomial (1 term), binomial (2 terms), trinomial (3 terms), polynomial (4+ terms).
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