Determine which of the following polynomials has (x + 1) a factor : (i) x 3 + x 2 + x + 1 (ii) x 4 + x 3 + x 2 + x + 1 (iii) x 4 + 3x 3 + 3x 2 + x + 1 (iv) x 3 - x 2 - (2 + )x +

(i) x 3 + x 2 + x + 1 ⇒ x + 1 = 0 ⇒ x = -1 p(x) = x 3 + x 2 + x + 1 p(-1) = (-1) 3 + (-1) 2 + (-1) + 1 = -1 + 1 -1 + 1 = 0 Remainder is zero (0), so (x + 1) is factor of this polynomial. (ii) x 4 + x 3 + x 2 + x + 1 ⇒ x + 1 = 0 ⇒ x = -1 p(x) = x 4 + x 3 + x 2 + x + 1 p(-1) = (-1) 4 + (-1) 3 + (-1) 2 + (-1) + 1 = 1 -1 + 1 -1 + 1 = 1 Remainder is not zero (0), so (x + 1) is not a factor of this polynomial. (iii) x 4 + 3x 3 + 3x 2 + x + 1 ⇒ x + 1 = 0 ⇒ x = -1 p(x) = x 4 + 3x 3 + 3x 2 + x + 1 p(-1) = (-1) 4 + 3 x (-1) 3 + 3 x (-1 2 ) + (-1) + 1 = 1 -3 + 3 -1 + 1 = 1 Remainder is not zero (0), so (x + 1) is not a factor of this polynomial. (iv) x 3 - x 2 - (2 + )x + ⇒ x + 1 = 0 ⇒ x = -1 p(x) = x 3 - x 2 - (2 + )x + p(-1) = (-1) 3 - (-1) 2 - (2 + )(-1) + = -1 - 1 + 2 + + = -2 + 2 + 2 = 2 Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.

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Mathematics

Determine which of the following polynomials has (x + 1) a factor : (i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1 (iii) x4 + 3x3 + 3x2 + x + 1 (iv) x3 - x2 - (2 + )x +

Polynomials

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Mathematics

Determine which of the following polynomials has (x + 1) a factor : (i) x<sup>3</sup> + x<sup>2</sup> + x + 1 (ii) x<sup>4</sup> + x<sup>3</sup> + x<sup>2</sup> + x + 1 (iii) x<sup>4</sup> + 3x<sup>3</sup> + 3x<sup>2</sup> + x + 1 (iv) x<sup>3</sup> - x<sup>2</sup> - (2 + )x +

Polynomials

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Find the zero of the polynomial in each of the following cases:
(i) p(x) = x + 5
(ii) p(x) = x - 5
(iii) p(x) = 2x + 5
(iv) p(x) = 3x - 2
(v) p(x) = 3x
(vi) p(x) = ax, a ≠ 0
(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers.

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x³ + x² - 2x - 1, g(x) = x + 1
(ii) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2
(iii) p(x) = x³ - 4x² + x + 6, g(x) = x - 3

Mathematics

Determine which of the following polynomials has (x + 1) a factor : (i) x<sup>3</sup> + x<sup>2</sup> + x + 1 (ii) x<sup>4</sup> + x<sup>3</sup> + x<sup>2</sup> + x + 1 (iii) x<sup>4</sup> + 3x<sup>3</sup> + 3x<sup>2</sup> + x + 1 (iv) x<sup>3</sup> - x<sup>2</sup> - (2 + )x +

Polynomials

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Find the zero of the polynomial in each of the following cases:(i) p(x) = x + 5(ii) p(x) = x - 5(iii) p(x) = 2x + 5(iv) p(x) = 3x - 2(v) p(x) = 3x(vi) p(x) = ax, a ≠ 0(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers.

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:(i) p(x) = 2x3 + x2 - 2x - 1, g(x) = x + 1(ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2(iii) p(x) = x3 - 4x2 + x + 6, g(x) = x - 3

Find the zero of the polynomial in each of the following cases:
(i) p(x) = x + 5
(ii) p(x) = x - 5
(iii) p(x) = 2x + 5
(iv) p(x) = 3x - 2
(v) p(x) = 3x
(vi) p(x) = ax, a ≠ 0
(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers.

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x³ + x² - 2x - 1, g(x) = x + 1
(ii) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2
(iii) p(x) = x³ - 4x² + x + 6, g(x) = x - 3