Equivalent expressions：練習問題

Which of the following is equivalent to the expression \( (4x^2 + 7x - 5) - (2x^2 - 3x + 8) \)?

Which of the following is equivalent to the expression \( \frac{x^2 - 9}{x + 3} \) for all \( x \neq -3 \)?

The expression \( \frac{2x + 5}{x - 3} \) can be rewritten in the equivalent form \( a + \frac{b}{x - 3} \), where \( a \) and \( b \) are constants. What is the value of \( a + b \)?

Which of the following is equivalent to the expression \( 3(x^2 - 2x) - 2(x^2 - 5) \)?

Which of the following is equivalent to \( \frac{2}{x+3} + \frac{3}{x-1} \) for all \( x \neq -3 \) and \( x \neq 1 \)?

The expression \( (3x + 4)^2 - (3x - 4)^2 \) is equivalent to \( ax \) for all values of \( x \). What is the value of the constant \( a \)?

The expression \( (ax^2 + 10)(3x - 4) - 5x^2 \) is equivalent to \( 12x^3 - 21x^2 + 30x - 40 \), where \( a \) is a constant. What is the value of \( a \)?

Consider the algebraic expression \( P = \frac{x^2 - 16}{x - 4} + 2(x + 3) - (x - 1) \), where \( x \neq 4 \).
a) Simplify the rational term \( \frac{x^2 - 16}{x - 4} \) by factoring the numerator.
b) Expand and distribute the terms in \( 2(x + 3) \) and \( -(x - 1) \).
c) Combine all terms to write the expression in the simplified form \( mx + b \).
d) What is the value of the expression when \( x = 5 \)?