If px<sup>4</sup> + qx<sup>3</sup> + rx<sup>2</sup> + sx + t = <math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='|' close='|'><mtable><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mtd><mtd><mi>x</mi><mo>-</mo><mn>1</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mn>2</mn><mo>-</mo><mi>x</mi></mtd><mtd><mi>x</mi><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>-</mo><mn>3</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>4</mn></mtd><mtd><mn>3</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math> then t is equal to -
Step-by-Step Solution
Key Concept: The constant term t in a polynomial f(x) is equal to f(0). By setting x=0 in the given determinant, we can find the value of t.
Given f(x) = px<sup>4</sup> + qx<sup>3</sup> + rx<sup>2</sup> + sx + t. To find t, put x = 0. Then t = f(0) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='|' close='|'><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>. Expanding along the first row: t = 0(0 - (-12)) - (-1)(0 - 9) + 3(4 - (-6)) = 0 + 1(-9) + 3(10) = -9 + 30 = 21.
Correct Answer: C