Polynomials：練習問題

Which of the following pairs of terms are unlike terms?

Find the coefficient of the \( x^2 \) term in the simplified expansion of the polynomial \( (3x^2 - x + 4)(2x - 3) \).

Factorise the polynomial expression \( 4x^2 - 12xy + 9y^2 - 25 \) completely.

Expand and simplify the expression: \(3y(y^2 + 2y - 5)\).

Simplify the expression \( (5a^2 - 2ab + 3b^2) - (2a^2 + 3ab - b^2) \).

Simplify by combining like terms: \( 6xy^2 + 3xy^2 - 2x^2y \).

Find the coefficient of the \( x^2 \) term in the expansion of the polynomial \( (2x - 3)(x^2 - 5x + 4) \).

In the figure, a rectangular metal sheet has outer dimensions \( (5x + 2) \) cm by \( (3x - 1) \) cm. If a rectangular hole of dimensions \( 4x \) cm by \( 2x \) cm is cut out from its centre, express the area of the remaining metal sheet as a simplified polynomial in terms of \( x \).

Consider the polynomial \(R(x) = 5x^3 - 9x + 2x^4 - 8 + 3x\).

(a) Simplify \(R(x)\) and write it in descending powers of \(x\).

(b) Determine the degree, the leading coefficient, and the constant term of the simplified polynomial \(R(x)\).

(c) If \(P(x) = x^4 - 3x^3 + 10\), find the polynomial \(S(x) = R(x) + P(x)\) and write the answer in descending powers of \(x\).

Consider the polynomials A and B, where:
A = \((3x - 2)(2x + 5)\)
B = \(5x^2 - 4x + 1\)

(a) Expand and simplify polynomial A.
(b) Subtract polynomial B from the simplified form of A.
(c) Find the value of the resulting expression from part (b) when \(x = -3\).