Factorise the following exprestions fully: 1.1.1 \( 4 x-x^{3}= \) \( 1.1 .2 \quad x^{4}+15 x^{3}-54 x^{3} \) \( 1.1 .3 \quad y-x y+x-1 \)
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Let’s break these down! For the first expression, \( 4x - x^3 \), you can factor out the common term \( x \), giving us \( x(4 - x^2) \). This can be further factored using the difference of squares, resulting in \( x(2 + x)(2 - x) \). For the second expression, \( x^4 + 15x^3 - 54x^2 \), we first factor out the greatest common factor, which is \( x^2 \). This simplifies to \( x^2(x^2 + 15x - 54) \). The quadratic can be factored further into \( (x + 18)(x - 3) \), leading to the final factorization of \( x^2(x + 18)(x - 3) \). Now for the last expression, \( y - xy + x - 1 \). You can rearrange and group it as \( (y - xy) + (x - 1) \), which can be further factored to \( y(1 - x) + (x - 1) \). By factoring out \( (1 - x) \), we get \( (1 - x)(y + 1) \). So, the factorizations are: 1. \( 4x - x^3 = x(2 + x)(2 - x) \) 2. \( x^4 + 15x^3 - 54x^2 = x^2(x + 18)(x - 3) \) 3. \( y - xy + x - 1 = (1 - x)(y + 1) \)
